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Origins of some Math terms
Math Words Alphabetical Index
Abscissa is the formal term for the x-coordinate of a point on a
coordinate graph. The abscissa of the point (3,5) is three. The
word is a conjunction of ab(from) + scindere (tear). Literally
then, abscissa is a line that has been cut or torn from another
line. The main root is closely related to the Latin root from which
we get the word scissors. I have a note that credits Leibniz with
the orinin of the term in 1692, but in 2006 I received a note from
Professor Barney Hughes that, "Fibonacci used the word in our
meaning several times in his book, De practica geometrie. " .
I read a blog (ok, I read it because it mentioned me..in fact,
misquoted me, but there was still some new stuff there, so : They
quoted Jeff Miller's site (always a good bet to be very accurate)
as saying that the first use was in 1659 in Miscellaneum
Hyperbolicum, et Parabolicum by Stephano degli Angeli. The site
also quoted some translation of the usage by Fibonacci, and the
usage seemed distinctly different than the present usage.. here is
one for you to decide. "Not to be overlooked is to show how to find
the square on line eb called the residue, recisum, or abscissa. It
is the difference between two lines commensurable only in their
squares, such as between lines ae and ab. For example, let ae be
the root of the rational number 720 and ab the number 10. Because
line ae was divided into two parts at point b, the squares on lines
ae and ab equal twice the product of ab by ae and the square on
line eb, as was shown above. Therefore subtract twice the product
of ab and ae from the squares on lines ae and ab; that is, subtract
20 times the root of 720 from 820. Now 20 roots of 720 equal the
root of 288000, the number arising from the product of 400 the
square of 20 and 720. The residue then is 820 less the root of
288000"
Absolute Value The word absolute is from a variant of absolve
and has a meaning related to free from restriction or condition.
The first use of "absolute value" in English seems to have been to
apply to real values. Jeff Miller's
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website on the Earliest Known Uses of Some of the Words of
Mathematics says," Absolute value is found in English in 1850 in
The elements of analytical geometry; comprehending the doctrine of
the conic sections, and the general theory of curves and surfaces
of the second order by John Radford Young (1799-1885): "we have AF
the positive value of x equal to BA - BF, and for the negative
value, BF must exceed BA, that is, F must be on the other side of
A, as at F', hence making AF' equal to the absolute value of the
negative root of the equation" [University of Michigan Digital
Library]." [See the page here] In 1876 Karl Weierstrass applied the
term to magnitude of complex numbers. From Miller's site again we
find "Absolute value was coined in German as absoluten Betrag by
Karl Weierstrass (1815-1897), who wrote: Ich bezeichne den
absoluten Betrag einer complex Groesse x mit |x|. [I denote the
absolute value of complex number x by |x|.]"
In "The Words of Mathematics", Steven Schwartzman suggests that
the use of the word for real values only became common in the
middle of the 20th century. This may be true, but the use for
signed numbers also appears in 1889 by Wentworth according to
Miller; "In 1889, Elements of Algebra by G. A. Wentworth has:
'Every algebraic number, as +4 or -4, consists of a sign + or - and
the absolute value of the number; in this case 4.' " (above). In
the 1893 edition of the same book he uses the term again, as shown
below, without any symbol.
The revision of Hall and Knight's Algebra, for Colleges and
Schools {"Revised and Enlarged for the use of American Schools"} by
F. L. Sevenoak in 1905 also uses the term without a sign. By 1934,
the word is still used without symbol in Walter W. Hart's
Progressive First Algebra,(pg 78), but in the 1939 edition of
College Algebra by Rosenbach
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and Whitman, the symbol is used as shown below
The symbol for absolute value is usually a pair of vertical
lines containing the number, as created by Weierstrass in 1876 (see
above). |3| is read as "The absolute value of three". The absolute
value of a real number is its distance from zero, so |3| = |-3| =
3. In words that says that the absolute value of three is equal to
the absolute value of -3 , and that both have a value of three.
For complex numbers the absolute value is also called magnitude
or length of the complex number. Complex numbers are sometimes
drawn as a vector using an Argand Diagram, and the length of the
vector Z=a+bi is |a+bi|. Stated another
way, the value of |a+bi|=
A symbol for the Absolute Difference of two numbers, or the
absolute value of the difference was created by Oughtred around
1630. Miller writes, "The tilde was introduced for this purpose by
William Oughtred (1574-1660) in the Clavis Mathematicae (Key to
Mathematics), composed about 1628 and published in London in 1631,
according to Smith, who shows a reversed tilde (Smith 1958, page
394)." I have seen this symbol used in an American text as late as
1893 when Irving Stringham used it in his list of symbols in
Uniplaner Algebra. The symbol seems no longer to be common in basic
maths classes in the US or in England today. After posting a
request for information to the Historia Matematica discussion group
about the use of the tilde to indicate absolute difference in
England I received the following update from Herbert Prinz:
"In modern English texts on navigation, nautical astronomy or
its history, the tilde is frequently used to express the function |
a - b |, where |x| stands for absolute value. E. g. Cotter, The
Complete Nautical Astronomer, 1969. I am not sure when this
practice started. In older texts on the same subject, say, Moore,
The Practical Navigator, 1800, one does not find the tilde used in
this way. For one, because
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instructions were given mostly verbally without the use of any
symbols at all. And second, the distinction from '-' was
unnecessary, as it was always understood, if not explicitly stated,
that one must subtract the smaller
In England the absolute value is often referred to as the
modulus function, and the two bars that make up the symbol are
sometimes called "modulus signs" according to a note posted by
Vicky Neale on the Ask NRich math site. The term modulus is used
both in America and England to represent the magnitude or length of
a complex number. The term is also used in a number of other
specialty ways in mathematics, the best known being the "congruence
modulus". The modulus of a congruence, often shortened to "mod" is
the base value with which the congruence is computed. We say A is
Congruent to B modulus C, if A divided by C and B divided by C have
the same remainder. C is called the modulus of congruence. It would
be written AB [mod C]
Modulus comes almost unchanged from the Latin from the
diminutive of modus (measure or amount), modulus for a small
measure. Vicky also pointed out that at one time the term was used
for, "A unit of payment used at Trinity College.... Fellows
received some number of moduli". Ms Neale also said she was
unfamiliar with the use of the ~ for absolute difference.
It was Gauss, Disquisitiones arithmeticae in 1801, who
introduced the term modulus of congruence, and the abreviation,
"mod". Cajori credits Jean Argand for the first use of modulus for
the length of a vector in 1814. I am not sure when the British
public schools started to use the term for the absolute value of a
number, and would love to know if someone has old books with these
terms (or others for the same idea).
Acute is from the Latin word acus for needle, with derivatives
generalizing to anything pointed or sharp. The root persists in the
words acid (sharp taste), acupuncture (to treat with needles) and
acumen (mentally sharp). An acute angle then, is one which is sharp
or pointed. In mathematics we define an acute angle as one which
has a measure of less than 90o.
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The first use of the term in Enlish was in Henry Billingsley's
translation of the Elements of Euclid. "An acute angle is that,
which is lesse then a right angle"; "an obtuse angle is that which
is greater then a right angle" .
Algebra comes from a book written in Arabic that revolutionized
how mathematics was done in western cultures. "Al-jebr
w'al-mugabalah" written by Abu Ja'far Ben Musa (about 825 AD) who
was also known as al-Khowarizmi. He is as famous among Arabs as
Euclid and Aristotle are to the Western World. He was probably the
greatest living mathematician of his period. The phrase Al-jebr at
the start of the title became the word Algebra in western
languages. The phrase loosly translated means "the reunion of
broken parts". Later, in medieval Europe, "algebrista" was became a
term for the person who set bones (the reunion of broken parts) and
since it was the barbers who did the bonesetting and blood-letting,
they were called an "algebrista".
Abu Ja'far Ben Musa is often mistakenly listed as an Arab
mathematician, but was in fact Persian, and Khowarizmi refers to
the area which was his home. Modern scholars believe he was born
near the Aral sea in what is now Turkestan. The literal translation
of his name means "father of Jafar and Son of Musa, from
Khowarizmi."
The first use of the word "algebra" in English was by the Welsh
mathematician and textbook writer, Robert Recorde in his Pathway of
Knowledge written about 1550.
Abu Ja'far Ben Musa was also the source of the word algorithm
(see below). His book, above, also includes the first use of what
we would today call the quadratic formula; although his description
was verbal and not in modern mathematical notation.
Algorithm, as it is used in mathematics means a systematic
procedure to solve a problem. The word is derived from the name of
the Persian mathematician, al-Khowarazmi (See algebra). The first
use of the word I am aware of was by G W Liebniz in the late
1600.
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Julio Gonzalez Cabillon posted the notes below in summarizing
earlier posts at Historia Matematica
I checked out the Latin of this Leibniz' first published account
of the calculus [_Acta Eruditorum_, vol. 3, pp. 467-473, October
1684], and I certainly find the word "Algorithmo". [*] "Nova
Methodvs pro maximis et minimis, itemque tangentibus, quae nec
fractas, nec irrationales quantitates moratur, & singulare pro
illis calculi genus, per G.G.L." (Leibniz's initials in Latin): On
page 469, Leibniz states: "Ex cognito hoc velut *Algorithmo*, ut
ita dicam, calculi hujus, quem voco *differentialem*, omnes aliae
aequationes differentiales inveniri possunt per calculum communem,
maximae que & minimae, item que tangentes haberi, ita ut opus
non sit tolli fractas aut irrationales, aut alia vincula, quod
tamen faciendum fuit secundum Methodos hactenus editas." A few
comments: 1. Both terms "Algorithmo" and "differentialem" are
italicised in the original. This must be emphasized, since either
Smith or his editor overlooked this 'petit' detail in "A Source
Book in Mathematics". 2. Please note that in the quoted passage,
Leibniz employs "algorithm" (in the sense of a systematic technique
for solving a problem) with a meaning that may suggest a new term
-- the context, and the italics conveys that possibility. 3.
Apparently, the first English translation of Leibniz' "Nova
Methodvs pro maximis et minimis..." was carried out by Joseph
Raphson in "The Theory of Fluxions, Shewing in a compedious manner
The first Rise of, and various Improvements made in that
Incomparable Method", London, 1715. Thereby, most probably the
earliest printed appearance, in English, of the term ALGORITHM (in
the sense of a systematic technique for solving a problem) is in
that treatise.
Algorithm remained a little known and little used term in
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western mathematics until the Russian mathematician Andrei
Markov (1903-????) introduced it. The term became very popular in
the areas of math focused on computing and computation.
Analogy The word analogy comes from the early Greek roots ana +
logos . Logos was the early Greek root for lots of related mental
constructions such as word, speech, logic, and reason. An analogy
refers to things that share a similar relation. Originally it was
more of a mathematical term interchangeable with ratio or
proportion; as in "2,4,8 is analogous to 3,6,12". Later this idea
of similar relations was extended to things that shared a logical
relationship. Analog clocks and computers are so named because they
operate off mechanical objects (gears and pulleys) that transform
motions in proportional movements.
Angle comes from the Latin root angulus, a sharp bend. As with
many g sounds the transfer from Latin to the German and English
languages switched to a k spelling. The word ankle is from the same
root. An angle is formed by two rays with a common endpoint.
The word Angles for the Germanic tribe that invaded England in
the 5th century, and from which words like Anglo-Saxon and English
are derived, was also drawn from the same root. "The Angles, says
the OED, are the people of Angul "a district of Holstein, so called
from its shape"; it goes on to say that Angul is the same word as
the Old English, Old Saxon and Old High German angul, a fish-hook -
which gives us the English word angling." [granthutchison, post on
Agora]
Apothem The distance from the center of a regular polygon to the
sides, the apothem, comes from the Greek term "to set off", as in
to set apart. The word is frequently pronounced "a poth' em' with
the accent on the second syllable, but the traditional, and
dictionary pronunciation is with the accent on the first syllable,
"ap' e thum" as in apogee, which
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shares the ap root, and means off from the Earth (gee from
geos). Apothem appears to be of modern origin despite its ancient
name, and seems to have first appeared in English in the mid
1800's
According to Jeff Miller's website on the first use of math
terms:
APOTHEM is found in 1828 in Elements of Geometry and
Trigonometry (1832) by David Brewster (a translation of Legendre):
The radius OT of the inscribed circle is nothing else than the
perpendicular let fall from the centre on one of the sides: it is
sometimes named the apothem of the polygon.
Are An are is a unit of measure for area equal to 100 square
meters. The word, and the unit of measure, seems to have been
created by the French and derived from the Latin word area with its
current meaning. The are is seldom used today, but its derivative
form, the hectare, is still a common unit of land measure in some
countries.
Arithmetic was the Greek word for number, and is closely related
to the root of reckon, which is becoming an obsolete term for count
(except in some parts of the western and southern US where they
"reckon" almost anything). . . . (that was a joke folks). In the
middle ages the best mathematicians of Germany were called
Reichenmeister and their arithmetic texts were reichenbucher The
beginning of the word is drawn from the Indo-European root ar which
is related to "fitting together" and gives us words like army, and
art. Order, adorn, and rate all come from variants of the same
root.
The first arithmetic book published in North America was Sumario
compendioso de las quentas de plata y oro que in los
reynos del Piru son necessarias a los mercaderes y todo
genero de tratantes Los algunas reglas tocantes al
Arithmetica. The title translates to "Comprehensive Summary of
the counting of silver and gold, which, in the kingdoms of Peru,
are necessary for merchants and all kinds of traders". The author
was Brother Juan Diez, a priest who
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arrived in Mexico with Cortez in 1519. The following is clipped
from an article, THE SUMARIO COMPENDIOSO: THE NEW WORLD'S FIRST
MATHEMATICS BOOK , in the Mathematics Teacher in February of 2001
by Shirley Gray and C. Edward Sandifer.
The author, Brother Juan Diez, arrived in Mexico with Cortez in
1519. In 1536, a printing press was set up in Mexico City, and the
following year, it went into operation and was used for printing
religious books. In 1556, the Sumario became the first book that
was not a religious book, and the twenty-fifth book of any kind, to
be published in the New World. The publication date of 1556 is
remarkable. It was long before any settlement in Jamestown (1607),
Plymouth Colony (1620), or Quebec City (1608). The New World's
first mathematics book in English was not published until 1703. A
Dutch mathematics book was published in 1730; a German book, in
1742; a French book, in 1775; a Portuguese one, in 1813; a Hawaiian
one, in 1833; and a mathematics book in Choctaw in 1835. Of all the
colonial mathematics books, the ones in Spanish are the most
interesting because they were mostly written in America for use by
people living in America. Books from the other colonies were mostly
American editions of European books or else were closely based on
European editions.
According to Bruce Burdick of Roger Williams University, "The
New World's first printed arithmetic (as opposed to a book, like
the Sumario Compendioso, that contains arithmetic or algebra but
whose main theme was something else) was the Arte para Aprender by
Pedro de Paz (Mexico, 1623)." For those seeking more information
about early Spanish math books in the Americas, Professor Sandifer
has an article about "mathematics books published in the Spanish
American colonies before 1700" and another article on the Breve
Arithmetica on his web page.
Associative The root of the word associative, is the Greek root
for our word social, soci. The first use of the word in the sense
of a mathematical property was probably by W R Hamilton around
1850.
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Association in mathematics refers to changing the grouping of
objects to be operated upon first. Since addition and
multiplication are binary operations (they work with two numbers at
a time) if we wish to add three numbers we have to choose which two
to add first. The associative property of addition says that in
adding 2+4+7, the same result will occur if we add 2+4 first, and
then add the result to 7 as would occur if we added 4+7 first, and
then added 2 + (the answer to 4+7). Formally the distributive
property of addition is written (A+B)+C= A+(B+C). There is an
identical property for multiplication.
Asymptote The asymptote of a function as it is now used is a
much narrower definition than the original Greek meaning. The word
joins the roots a (not), with sum (together) + piptein (to fall)
and literally means "not falling together", or not meeting. The
word is believed to have been known to Apollonius of Perga before
200 BC. Originally it was used for any two curves that did not
intersect. Proclus writes about both asymptotic lines, and
symptotic lines (those that do cross). Now symptotic is almost
never heard, and asymptote is used primarily for straight lines
that serve as a limiting barrier for some curve as one of its
parameters approaches infinity (+/-).
The ~ symbol is often used to indicate that one function is
asymptotic to another. One might write f(x)~ g(x) if the ratio of
f(x) and g(x) approach 1 as x -> infinity.
The pet base of the root piptein gives us words like petition,
petal, petite and propitious.
Average The meaning of average, as it is used in math today,
comes from a commercial practice of the shipping age. The root,
aver, means to declare, and the shippers of goods would declare the
value of their goods. When the goods were sold, a deduction was
made from each persons share, based on their declared value, for a
portion of the loss, their AVERAGE.
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In response to a question about the use of the x-bar
symbol, , for averge(mean) value of a sample, John Harper of
Victoria University in New Zealand sent a response including the
following information:
> R.A. Fisher used that notation, in "On an absolute
criterion for fitting frequency curves Messenger of Mathematics",
v. 41: 155-160 (1912) on p.157. (Univ of Adelaide has put Fisher's
collected works on the Web) I don't know if he was the first. John
Harper, School of Mathematical and Computing Sciences, Victoria
University, PO Box 600, Wellington, New Zealand
Jeff Miller's web page provides some additional material:
for the sample mean. This usage derives from the practice of
applied mathematicians of representing any kind of average by a
bar. J. Clerk Maxwell's "On the Dynamical Theory of Gases
(Philosophical Transactions of the Royal Society, 157, (1867) p.
64) uses v-bar for the "mean velocity" of molecules while W.
Thomson & P. G. Tait's
Treatise on Natural Philosophy (1879) uses for the centre of
inertia, wx / x. Karl Pearson, the leading statistician of the
early 20th century, was from such a physics background. Pearson and
his contemporaries used the bar for sample averages and for
expected values but eventually E
replaced it in the latter role. The survival of for the sample
mean is probably due to the influential example of R. A. Fisher who
used it in all his works from the first, " On an Absolute Criterion
for Fitting Frequency Curves," (1912).
Billion seems to have been a French creation, and was originally
bi-million. The term originally meant 10^12 or one million
millions, and still has this meaning in many countries today. In
the US and some other countries it is used for 10^9 or one thousand
million. The table below compares the names as used in the US and
in Germany: Value -----German name--------US name 10^6 -----
Million ---------- Million 10^9 ------
Millard------------Billion
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10^12 ----- Billion -----------Trillion 10^15------ Billiarde
-------- Quadrillion
Cajori attributes the first publication of the words above
million to Nicholas Chuquet. Here is a quote from his A History of
Elementary Mathematics with Hints on Methods of Teaching:
Their origin dates back almost to the time when the word million
was first used. So far as known, they first occur in a manuscript
work on arithmetic by that gifted French physician of Lyons,
Nicolas Chuquet He employs the words byllion, tryllion,
quadrillion, quyllion, sixlion, septyllion, octyllion, nonyllion,
"et ainsi des aultres se plus oultre on voulait proceder" to denote
the second, third, etc. powers of a million, i.e. (1,000,000)2,
(1,000,OO0)3, etc. Evidently Chuquet had solved the difficult
question of numeration. The new words used by him appear in 1520 in
the printed work of La Roche. Thus the great honour of having
simplified numeration of large numbers appears to belong to the
French. In England and Germany the new nomenclature was not
introduced until about a century and a half later. In England the
words billion, trillion, etc., were new when Locke wrote, about
1687. In Germany these new terms appear for the first time in 1681
in a work by Heckenberg of Hanover, but they did not come into
general use before the eighteenth century. About the middle of the
seventeenth century it became the custom in France to divide
numbers into periods of three digits, instead of six, and to assign
to the word billion, in place of the old meaning, (1000,000)2 or
1012, the new meaning of 109
In The Book of Numbers by John Conway and Richard Guy (pp.
14-15) they write
These arithmeticians [Chuquet and de la Roche] used "illion"
after the prefixes b, tr, quadr, quint, sext, sept, oct and non to
denote the 2nd, 3rd, 4th, 5th, 6th, 7th, 8th and 9th powers of a
million. But around the middle of the 17th century, some other
French arithmeticians used them instead for the 3rd, 4th, 5th, 6th,
7th, 8th, 9th and 10th powers of a thousand. Although condemned by
the greatest lexicographers as "erroneous" (Litr'e) and "an entire
perversion of the original nomenclature of Chuquet and de la Roche"
(Murray), the newer usage is now standard in the U.S., although the
older one survives in Britain and is still standard in the
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continental countries (but the French spelling is nowadays
"llon" rather than "llion". Because of continued conflict with
England for the first fifty years of the new United States
existance, it was much more willing to base the foundation for its
numeration system on the method of the French, who had supported
them in their revolution. In spite of this, "In many textbooks
prior to the War of 1812 (eg. those by Consider and John Stery
1790, John Vinall 1792, and Johann Ritter 1807) if any numbers
higher than 999,999,999 were discussed, the British system was
used." [for example 1,000,000,000 was one-thousand million rather
than one-billion ] {from Karen D. Michalowicz and Arthur C Howard
in "Pedagogy in Text", from the NCTM's A History of School
Mathemaitics}
Cardinal numbers are numbers that express amounts, as opposed to
ordinal numbers, which express order or rank. The term is from the
Latin, cardin, for stem or hinge. Cardinal today means most
important or principal, with other things depending (hinging) on
it. The first use appears to have been by R Percival in 1591,
Cardioid The path of a point on a circle as it rolls around
another circle of the same size is sort of heart shaped and thus
the term is from the Greek root for heart, kardia.
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Here is a note on the origin of the term from a post by Julio
Gonzalez Cabillon: CARDIOID was first used by Giovanni Francesco
Mauro Melchior Salvemini de Castillon in "De curva cardiode" in the
Philosophical Transactions of the Royal Society (1741). Giovanni
Castillon was born on January 15, 1708, in Castiglione (hence his
name), and died on October 11, 1791, in Berlin. I've taken his
dates from Poggendorff's _Bibl.-lit. Handwoerterbuch_.
A nice animation of the generation of a cardiod by a circle
rolling upon it is given at the Mathworld page.
The cardiod is a degenerate form of a limacon. The Polar
equation of a limacon is r = b + a Cos(t). If b is smaller than a
then the limacon will have an internal loop. If b is larger than a,
but smaller than 2a, then the limacon will have a concave "dimple".
if b is greater than 2a then the limacon is convex. When b=a, the
shape is a cardiod.
At Jeff Miller's web site on the first use of math words I
found,
The term LIMAON was coined in 1650 by Gilles Persone de Roberval
(1602-1675) (Encyclopaedia Britannica, article: "Geometry"). It is
sometimes called Pascal's limaon, for tienne Pascal (1588?-1651),
the first person to study it. Boyer (page 395) writes that "on the
suggestion of Roberval" the curve is named for Pascal.
Center The word center comes to us from a Greek root, kentrus,
for a spur or sharp pointed object. The relation to the center of a
circle seems obvious. A sharp point was made at a center to fix the
spot, and a more dull object was dragged around the center to form
the circle.
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Century Although now used almost exclusively for a period of one
hundred years, century was originally the Latin term for any
collection of one hundred items. In the Roman army a company
consisted of one hundred men, and each was called a centurion.
Cevian A word created by French geometers around the end of the
19th century to honor the Italian Giovanni Ceva (1650?-1735). A
cevian is a line segment from a vertex of a triangle to a point on
the opposite side. The median, altitude, and angle bisector are all
examples of cevians. The perpendicular bisector, in most cases, is
not a cevian because it does not pass through a vertex of the
triangle. Note that a cevian, may cut the opposite side outside the
triangle.
Julio Gonzalez Cabillon wrote, "the French word CEVIENNE, which
was proposed by Professor A. Poulain (Faculte catholique d'Angers,
France) in 1888. Naturally, he derived the word from the surname of
the Italian mathematician Giovanni Ceva (1647?-1734)."
Chaos Although the ideas of chaos theory as we know it today
have been actively studied at some level for most of the 20th
century, the word as a mathematical term dates only from an article
in American Mathematical Monthly in 1975, "Period Three Implies
Chaos". The Greek root khaox was for an empty space. This meaning
still persists in archaic usage where it refers to a canyon or
abyss. The evolution of the word to mean disorder seems to come
from reference to the time before God created the universe. The
empty space was with out order and the creation filled the
emptiness and created order.
A more common form of the word exists today, but few people are
aware of the connection. At the start of the 17th century, a
Flemish scientist named Jan Baptist van Helmont was studying the
bubbles that rise when fruit juice was allowed to stand. These
strange vapors, without shape or form, reminded him of the Greek
idea of Chaos, so he called
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them by the Germanic (Flemish is a dialect of German) spelling
of chaos, gas.
The physical objects formed out of the void were called the
cosmos, the Greeks word for orderly or well formed. Today we often
hear people refer to the Universe as the cosmos. When Robert
Milliken, the American physicist, sought a term for the radiation
that seemed to be coming from everywhere in the universe (the
cosmos) he suggested the name Cosmic Rays . Today the word cosmos
also remains as the root of words like cosmopolitan and
cosmetics.
Chi Square The statistical test, and the name for it are both
credited to Karl Pearson around the year 1900. The actual
distribution now called the Chi-Square distribution was discovered
earlier by Helment
The Chi_Square test is often used to assess the "goodness of
fit" between an obtained set of frequencies in a random sample and
what is expected under a given statistical hypothesis.
The distribution is named for the letter Chi,, the 22nd letter
of the Greek alphabet.
Chord The Greek root of the chord, chorde, means gut or string.
The musical use of the term comes from a contraction of accord, two
strings played together.
Circle The Latin root of the word circle is circus. The
traditional shape of the large roofless enclosures in which the
famous Roman Chariot races were run was circular or oblong, and
thus the word came to described this shape as well.
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Congruent The Latin word congruere meant "coming together" or
"working together". I learned from Glen Woodburn recently that,
"Actually, gruere comes from the latin word grui which means to be
in harmony with. So congruent translates to mean together in
harmony with." Whether applied to a geometric shape, or a military
unit, it meant that all the parts fit together. According to a
message from Nathan Sidoli, in Euclid's Elements the "word that
Heath translates as "coincides" is *efarmo^zein* - to fit
exactly"
. Nathan refers to Common Notion 4 in Book one, which Heath
translates as "Things which coincide with one another, are equal to
one another."
During the 16th century translations of Euclid into Latin began
to use the Latin term for Common Notion 4. In a note to the Math
Hisotry list J. Cabilon wrote that "Christoph Clavius (1537?-1612)
wrote: '...Hinc enim fit, ut aequalitas angulorum ejusdem generis
requirat eandem inclinationem linearum, ita ut lineae unius
conveniant omnino lineis alterius, si unus alteri superponatur. Ea
enim aequalia sunt, quae sibi mutuo congruunt.' (vol. I, p.
363)"[emphasis added]
At Jeff Miller's web site there are several notes on the
development of the term congruence. In particular he says that, "In
English, writers commonly refer to geometric figures as equal as
recently as the nineteenth century. In 1828, Elements of Geometry
and Trigonometry (1832) by David Brewster (a translation of
Legendre) has: Two triangles are equal, when an angle and the two
sides which contain it, in the one, are respectively equal to an
angle and the two sides which contain it, in the other."
The modern symbol for congruence common to most US high school
texts, which combines the tilde ~ above an equal
sign, =, was first used by many writers for similarity as well.
It is sometimes used with the wave inverted also.
Leibniz used a tilde with a single underline as a unique symbol
for congruence, but so many symbols were in use that it did not
catch on. According to Cajori, the use of the modern symbol for
congruence became the accepted practice around the beginning of the
20th century. He suggests the first use was by G. A. Hill and
George B Halstead. The symbol is still not universally accepted and
was not used in England at the time of his writing because of
confusion with the tilde symbols use for difference.
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I recently (Feb, 2004) posted a request for information to the
Historia Matematica discussion group and received the following
update on the use of the ~ symbol in England, and the related
question of a symbol for congruence. According to a post from
Herbert Prinz,
"In modern English texts on navigation, nautical astronomy or
its history, the tilde is frequently used to express the function |
a - b |, where |x| stands for absolute value. E. g. Cotter, The
Complete Nautical Astronomer, 1969. I am not sure when this
practice started. In older texts on the same subject, say, Moore,
The Practical Navigator, 1800, one does not find the tilde used in
this way. For one, because instructions were given mostly verbally
without the use of any symbols at all. And second, the distinction
from '-' was unnecessary, as it was always understood, if not
explicitly stated, that one must subtract the smaller from the
larger value." .
Tony Mann pointed out that in England the symbol was, "commonly
used for 'is isomorphic to', and is used colloquially for 'is
essentially equivalent to'." John Harper of Victoria University in
New Zealand added that, "Geometric congruence was indicated by 3
parallel equal lines: , an equals sign with a straight underline.
It still is, according to Borowski & Borwein "Collins
Dictionary of Mathematics" (HarperCollins, Great Britain 1989) who
give ~ on top of = only for approximate equality." Cajori credits
the creation of the symbol for geometric congruence to Reimann and
was used by Bolyai.
Gauss used the term congruent in modular arithmetic to refer to
numbers which had the same remainder upon divison, for example 12 7
mod 5 since each has a remainder of two when divided by five.
Congruent Numbers was a brand new term to me when I read a neat
blog at Bit-Player about it after a recent news release from AIM
(American Institute of Mathematics) announced that all the
congruent numbers up to 1 trillion have been enumerated. Well, job
done I guess. The blog is so well written that I am not about to
try to replicate all that
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good work, go read it. If you want more, here is a link from the
AIM on the topic.
Conjugate is the union of the common Latin prefix com (together)
and the root juge (yoke) and means to bind together in a pair.
Mathematically it is often used for things that are opposites in
some way, as in the complex conjugates. The same word in grammar
refers to words of a common origin and related meaning, and in
biology to an act of sexual union, for which the more common term
is conjugal relations.
Converse is from the Latin roots com(great or intense) + vertere
(to turn). The literal meaning is "to turn away". The verb converse
(as in conversation), which has the same spelling, is from a
completely different root.
Dean The term now used for the head of a department or faculty
at a college is derived from the Latin deaconus which meant "chief
of ten". The similar sounding deacon, for a church leader, is not
related and comes from the Greek root diakonos for a servant.
According to John Conway, the literal meaning is "one who raises
the dust".
The Online Etymology Dictionary indicates that doyen comes from
the same term... "doyen 1422, from M.Fr. doyen "commander of ten,"
from O.Fr. deien (see dean).
Diagonal comes from the Greek roots dia( to pass through or
join) + gonus [angle] and describes the line segment which passes
from the vertex of one angle to another in a polygon. The word
diagonal was probably first used in a geometric sense by Heron of
Alexandria.
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Diagram joins the roots dia(to pass through or join) with gram
(written or drawn and earlier carved). It literally means "that
which is marked out", as by the crossing of two lines. This leads
me to wonder how old the expression, "X marks the spot", could
be.
Divide shares its major root with the word widow. The root vidua
refers to a separation. In widow the meaning is obvious, one who is
separated from the spouse. A similar version of the word was often
meant to describe the feeling of bereavement that a widow would
feel. The prefix, di, of divide is a contraction of dis, a two
based word meaning apart or away, as in the process of division in
which equal parts are separated from each other. Notice that the vi
part of vidua is also derived from a two word, and is the same root
as in vigesimal (two tens), for things related to twenty. An
individual is one who can not be divided.
In a division problem such as 24 / 6 = 4 the number being
divided, in this case the 24, is called the dividend and the number
that is being used to divide it, the 6, is called the divisor. The
four is called the quotient. If the quotient is not a factor of the
dividend, then some quantity will remain after division. This
quantity is usually called the remainder, although residue
sometimes is used. The Treviso Arithmetic uses the word lauanzo for
remainder. In Frank Swetz's book, Capitalism and Arithmetic he
gives, "The term lauanzo apparently evolved from l'avenzo, meaning
a surplus, or in a business context, a profit." Swetz also points
out that in the 15th Century the term partition (partire in Latin)
was synonymous with the word divisision
In today's schools almost every grade school student learns to
divide, so students may be surprised to learn that in the 16th
century schools Division was only taught in the University. One of
the first arithmetics for the general public that treated the
subject of division was Rechenung nach der lenge, auff den Linihen
vnd Feder by Adam Riese. Here is how the Math History page at St
Andrews University in Scotland described it,
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"It was published in 1550 and was a textbook written for
everyone, not just for scientists and engineers. The book contains
addition, subtraction, multiplication and, very surprisingly for
that period, also division. At that time division could only be
learnt at the University of Altdorf (near Nrnberg) and even most
scientists did not know how to divide; so it is astonishing that
Ries explained it in a textbook designed for everyone to use." I
think it is even more astonishing that the sitution described still
existed in 1550 in Germany. Perhaps the earliest "arithmetic" to
provide instruction in the local vernacular of the common people
was the 1478 "Treviso Arithmetic", so named because it was printed
in the city of Treviso (the author is unknown) just north of
Venice. Frank J Swetz writes about the situation in Capitalism and
Arithmetic (pg 10): From the fourteenth century on, merchants from
the north travelled to Italy, particularly to Venice, to learn the
arte de mercadanta, the mercantile art, of the Italians. Sons of
German businessman flocked to Venice to study...
Early algorithms for division: By the middle ages there seem to
have been five approaches to the process of division.
The first was called the Galley, galea, or Scratch method. This
method was efficient in a period of expensive paper and quill pens
since it required less figures than other methods. Even the modern
long division method requires more figures. The name Galley was
used because the resulting pattern after the division left a
picture that seemed to remind the early reckoning masters of the
shape of a ship at sail. The term scratch has to do with the
crossing out of values to be replaced with new ones in the process.
The ease with which this could be done on a sand board or counting
board made it a popular approach in the cultures of the East, and
the method is believed to come from the early Hindu or Chinese. For
example, Cajori writes, "It will be remembered that the scratch
method did not spring into existence in the form taught by the
writers of the sixteenth century. On the contrary, it is simply the
graphical representation of the method employed by the Hindus, who
calculated with a coarse pencil on a small dust-covered tablet. The
erasing of a figure by the Hindus is here represented by the
scratching of a figure." He also comments on the popularity of this
method, " For a long time the galley or scratch method was used
almost to the entire exclusion of the other methods. As late as the
seventeenth
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century it was preferred to the one now in vogue. It was adopted
in Spain, Germany, and England. It is found in the works of
Tonstall, Kecorde, Stifel, Stevin, Wallis, Napier, and Oughtred.
Not until the beginning of the eighteenth century was it superseded
in England. "
Here is an image comparing how the galley method works shown
beside the current US Model for long division, which the Italians
called a danda. The page the image is from has a nice step by step
illustration of the process. I have recently (2005) acquired a
German student "copy book" from 1804 which seems to show the Galley
division method and the student's illustration of the ship around
the work. (below right)
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A second method that was sometimes taught was the process of
repeated subtraction. The image below shows an example from a
popular Arithmetic in the US by Charles Davies, published in 1833.
I have seen this method in an English textbook as late as 1961
(Public School Arithmetic by Baker and Bourne). It also appears in
a 1932 US publication of Practical Arithmetic, by George H. Van
Tuyl, and perhaps in others
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A third method was called per repiego by parts, which I have
seen in books into the 20th century. In this method a division was
accomplished by breaking the divisor into its factors, and then
dividing the dividend by one of the factors, and sequentially
dividing the resulting quotient by each remaining factor in turn to
get a final quotient. The problem below is modeled on a problem in
the 1919 copyright A School Arithmetic, by Hall and Stevens.
divide 92467 by 168 or 4 x 6 x 7 4|92467 6|23116 . groups of
four and 3 units over 7| 3852 .. groups of 24 (4x6) and 4 foursover
___550 groups of 168 and 2 twenty-fours over The complete remainder
is 2 (24) + 4(4) + 3 = 67 This method was presented in Liber Abaci,
by Fibonacci in 1202. After introducing how to divide by numbers of
one digit, and then larger primes, he develops a set of
"Composition Rules" for numbers with more than one digit. A
composed fraction might look like . Fibonacci used the Arabic
method of writing fractions from right to left, and this composed
fraction would be read as 4/5 + 2/25 + 1/75; or in modern notation,
67/75 with each part of the numerator being read over the product
of all the denominators below or to the right. The "composition" of
75 would be a fraction with 1 0 0 and 3 5 5 in the denominator, the
fraction 1/75. When he divides 749 by 75, he first uses only the
first denominator, 3. The quotient of 749 by three is 249 with a
remainder of 2. The 2 is placed as a numerator over the three, and
the 249 is divided by the second number in the denominator (a
five). 249 divided by 5 gives 49 with a remainder of four. This
remainder, 4, is placed as a second number in the numerator over
the five in the denominator. Now the 49 is divided by
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the final number in the denominator (another five) and the
quotient is 9 with another remainder of four. This four is placed
over the final five and the nine is placed to the right as the
quotient. Fibonacci then gives the answer of
749 divided by 75 as which would be 9 and 4/5 + 4/25 + 2/75 or 9
74/75.
A fourth method, which is similar to what we would now called
short division except that the student used a table of division or
multiplication facts. The method was called per colona, by the
column, or per tavoletta by the table, in reference to the table of
facts used. An example of this method is shown below from another
popular American arithmetic by Nicholas Pike, from 1826. The use of
tables to aid in multiplication and division were a common practice
from the 1400s up to the early 20th century.
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The fifth is the true ancestor of the method most used for long
division in schools today, and was called a danda, "by giving". In
his Capitalism and Arithmetic, Frank J Swetz gives The rationale
for this term was explained by Cataneo (1546), who noted that
during the division process, after each subtraction of partial
products, another figure from the dividend is given to the
remainder. He also says that the first appearance in print of this
method was in an arithmetic book by Calandri in 1491. The method
was frequently called the Italian method even into the 20th century
(Public School Arithmetic, by Baker and Bourne, 1961) although
sometimes the term Italian method was used to describe a form of
long division in which the partial products are omitted by doing
the multiplication and subtraction in one step. The image below
shows a typical long division problem with the partial products
crossed out and the resulting "Italian method" on the right.
The early uses of this method tend to have the divisor on one
side of the dividend, and the quotient on the other as the work is
finished, as shown in the image below taken from the 1822 The
Common School Arithmetic : prepared for the use of academies and
common schools in the United States by Charles Davies. Swetz
suggests that it remained on the right by custom after the galley
method gave way to the Italian method in the 17th century. It was
only the advent of decimal division, he says, and the greater need
for alignment of decimal places, that the quotient was moved to
above the number to be divided.
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In a recent Greasham College lecture by Robin Wilson at
Barnard's Inn Hall in London, he credited the invention of the
modern long division process to Briggs, "The first Gresham
Professor of Geometry, in early 1597, was Henry Briggs, who
invented the method of long division that we all learnt at
school."
I recently found a site called The Algorithm Collection Project.
where the authors have tried to collect the long division process
as used by different cultures around the world. Very few of the
ones I saw actually put the quotient on top as American students
are usually taught. In one interesting note, a respondent from
Norway showed one method, then explained that s/he had been taught
another way, and then demonstrates the common American algorithm,
but adds a note that says, but no one is using this algorithm in
Norway anymore. I might point out that the colon, ":" seems to be
the division symbol of choice if this sample can be generalized as
it was used in Norway, Germany, Italy, and Denmark. The Spanish
example uses the obelisk, and the other three use a modification of
the "a danda" long division process. The method labled "Catalan" is
like the "Italian Method" shown above where the partial products
are omitted. (More about division symbols at Symbols of
Division
Dozen The word dozen is a contraction of the Latin Duodecim (two
+ ten). This root also appears in dodecagon (from duodecagon) and
duodenum, the first part of the intestine
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that is about twelve inches long. Some math and language
historians think that a dozen is one of the earliest primitive
groupings, perhaps because there are approximately a dozen cycles
of the moon in a cycle of the sun. It appears to be the basis of
many larger values that were developed by many cultures. A shock
was 60, or five dozen (a dozen for each finger on one hand) and
many cultures had a "great hundred" [see hundred] of 120 or ten
dozen (a dozen for each finger on both hands). The Romans used a
fraction system based on 12 and the smallest part, an uncil became
our word for an ounce. Charlemagne established a monetary system
that had a base of twelve and twenty and the remnants persist in
many places. In English money today 100 pence equals a Pound, but
only a few short years ago a Pound was divided into 20 shillings of
12 pence each.
First is a native English word from the Old English fyrst which
was a variant of fore (front)
Fraction comes from the Latin word frangere, to break. A
fraction, then, originally represented the broken portion of some
whole. The first known use of the word in English is by Geoffrey
Chaucer in 1391 in the work, A Treatise on the Astrolabe.
By the middle of the 19th Century fraction was used to describe
parts larger than the whole as well. In the 1876 edition of Davies'
Practical Arithmetic he lists as Article 114. "There are six kinds
of fractions:" He then goes on to define
"1. A Proper Fraction is one whose numerator is less than the
denominator" "2. An Improper Fraction is one whose numerator is
equal to, or exceeds the denominator." "3. A Simple Fraction is one
whose numerator and denominator are both whole numbers." (Note this
is not necessarily what modern teachers would call in "simplest
form", for example 8/4 is a simple fraction) "4. A Compound
Fraction is a fraction of a fraction or several fractions connected
by the word of or x. The
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following are compound fractions: 1/2 of 1/4, 1/3 of 1/3 of 1/3,
1/7 x 1/3 x 4." "5. A Mixed Number is a number expressed by an
integer and a fraction." "6. A Complex Fraction is one whose
numerator or denominator is fractional; or, in which both are
fractional," In the Fourth Yearbook of the NCTM in 1929 one of the
curriculum changes listed for the State of New York included in the
list for the 1910 syllabus, "Fractions, including complex fractions
of the 'apartment house' type." (page 161) I assume the "mixed
number over a mixed number" is the type of problem referred to, but
am still trying to find confirmation of this.
Many modern elementary teachers get upset by the use of the term
"reduce a fraction". I think this is mostly because they are not
familiar with the origin of the term and only understand the word
"reduce" to mean "make smaller", which is certainly one of the most
common definitions of the word in modern dictionaries. I hope the
the following will make them more understanding of those of us who
are VERY old, and still remember when the term had a broader
meaning.
According to the OED, the first use of the term in the sense of
reducing a fraction was in 1579 in a book by Thomas Digges.
Reduction is defined in the 1850 edition of Frederick Emerson's
North American Arithmetic, Part Third, for Advanced Scholars as
"the operation of changing any quantity from its number in one
denomination to its number in another denomination."(pg 29, see
image here) On the following page it asks the student to "reduce 7
bushels and 6 quarts to pints.". Later in the section on fractions
it defines, "Reduction of fractions consists in changing them from
one form to another, without altering their value." This broader
language is preserved in most later texts for the next seventy or
so years. It is defined in Milne's Progressive Arithmetic (1906,
William J Milne) thusly, "The process of changing the form of any
number without changing its value is called reduction." An almost
identical definition appears in Davies and Peck's 1877 Complete
Arithmetic, Theoretical and Practical(page 84, art. 66). All the
books include reduction of fractions to higher terms as well as
lower terms, and reduction of "decimals to common fractions".
In the Late 1930's and 40's arithmetic textbooks seemed to have
totally omitted the broader definition, and treat
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reduce as a vade mecam for fractions in "lowest terms" or
"simplest terms". In Learning Arithmetic (6) by Lennes, Rogers and
Traver, (1942) the term reduction appears in the index only as a
subheading under "fractions". The first occurance in the text, on
page 36, without prior definition introduces students to a set of
problems with the directions, "Reduce the fractions below to
simplest forms". In Making Sure of Arithmetic by Silver Burdett
(1955) the word "reduce" does not appear in the index at all, but
on page 8 it contains, "When the two terms of a fraction are
divided by the same number until there is no number by which both
terms can be divided evenly, the fraction is reduced to lowest
terms." [emphasis is from text]. By 1964, The Universal
Encyclopedia of Mathematics by Simon and Schuster contains "A
fraction is reduced, or cancelled, by dividing numerator and
denominator by the same number." (pg 364) Later on the same page
they note, "a fraction cannot be reduced if numerator and
denominator are mutually prime" indicating that when they said "the
same number" in the first statement, they meant a positive integer.
This definition leads to "reduction" of fractions as making the
numerator and denominator both smaller.
The roots of the word reduce are from the Latin re for back or
again, and dicere which means "to lead". The latter root is also
found in the word educare which is literally, to lead out, and is
the source of our modern English word, educate.
Frustum (sometimes spelled frustrum) is from the Latin and means
"a piece broken off". Mathematically it usually refers to a part of
a solid cut off between two parallel planes, as opposed to
truncated. The Indo-European root of frustum is bhreus and is
related to cutting, crushing, or pounding. Related words from the
same root are fragment, bruise, and possibly brush (from a bundle
of cut twigs).
Jeff Miller's web site on the earliest use of math words
includes a note on how frequently the term is misspelled as
"frustrum".
"This word is commonly misspelled as "frustrum" in, for example,
Samuel Johnson's abridged 1843 Edition of his
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dictionary. The word is spelled correctly in the "Frustum" entry
and the "Hydrography" entry in the 1857 Mathematical Dictionary and
Cyclopedia of Mathematical Science, but it is misspelled in the
entry "Altitude of a Frustrum." The word is misspelled in the 1962
Crescent Dictionary of Mathematics and remains misspelled in the
1989 Webster's New World Dictionary of Mathematics, which is a
revision of the Crescent dictionary. The word is also misspelled in
at least three places in The History of Mathematics: An
Introduction (1988) by David M. Burton. This has become so common
it may almost be considered an alternative spelling. He also has,
of course, a notation on the first use of the term in English,
"FRUSTUM first appears in English in 1658 in The Garden of Cyrus or
the Quincuncial Lozenge, or Net-work Plantations of the Ancients
... Considered by Sir Thomas Brown: "In the parts thereof [plants]
we finde..frustums of Archimedes" (OED2)."
Geometry is derived from the conjunction of the Greek word for
the Earth, Geos, and the term for "to measure", metros. Literally
then, Geometry means "to measure the Earth". According to the Greek
legends the first things created out of the "chaos" were the earth,
Gaia and the sky, Ouranos (which would become Uranus in Latin). The
Greek word Gaia was a name not only for the earth, but for the
goddess of the Earth. Although the Greeks and the Latins pronounced
the word Gay' yuh it came into English pronounced more like Jee' uh
from which we get the many Geos rooted words such as geography and
geology.
I recently read a post by G.L.Narasimham to a geometry
discussion list that pointed out that the common term in some
Indian dialects, was very closly related
A commercial (advertisement for onion soup spice, CBR Masala )
in Hindu newspaper in India a couple of years ago mentioned origin
of the word Geometry.As it is intersting, I reproduce it below, not
verbatim: --- Geometry ( Jyo-Miti) Kalpasutra is an important
source of Vedic mathematics. The sections called sulba-sutras deal
with measurement and construction of such Yagna-vedis (platform for
religious rites) that involved geometrical propositions and
problems
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related to geometrical figures. The word sulba means a cord,
rope, filament or string and the word root means 'measuring'. It is
interesting to note that among the Egyptians, geometry of surveying
was considered to be the science of rope structures (harpendona
'ptae) !! .. They thus appear to be the Egyptian counterparts of
Indian sulbavids. Some 50 years ago text books on geometry in Tamil
language were entitled 'Jyomiti Ganitam' The word geometry owes its
origin to India, Jya=Earth. Miti = Measure. Hence 'Jyamit' meaning
measurement of earth or figures drawn on earth, gave rise to the
present term geometry. Courtesy Dr. V.S. Narasimhan, Chennai. 'A
Concise History of Science of India', Indian National Science
Academy, pp 139,149. -------
Just as the study of the Earth, Geology, recalls the ancient
Greek goddess of the earth, the term Uranology is the study of the
sky, but the more common term today is astronomy. But do not
despair for the lost memory of the Greek god of the sky, for he is
preserved in the name of the planet discovered in 1781 by William
Herschel. Herschel thought he had discovered a comet and published
the result for the Royal Society. Although Herschel wanted to name
the planet for King George III, remembered as the bad guy in the
American revolution by U.S. History; Johann Bode (see Bode's Law)
came up with the suggestion for the name that stuck, Uranus.
Hershcell continued to call the planet Georgium Sidus, the Georgian
Star after his royal patron.
Uranus, the sky god, is also remembered through a discovery a
few years later (1798) by chemist Martin Klaproth. It was a
tradition of chemists to name metals after planets, so Klaproth
named his new metal after the new planet, calling it Uranium.
Strangely, he later discovered another new metal and decided to
name it after the earth, but instead of using the Greek goddess of
earth, he chose the Roman equivalent and called his new metal,
tellurium. The Roman goddess of the earth was known by two names.
The first was terra, which gives us words like terrestrial, and its
better known opposite, extra-terrestrial. The other was tellus,
which is almost non-existent today, except in the name of
Klaproth's metallic discovery.
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Googol A number invented by Milton Sirotta, the eight year old
nephew of Dr Edward Kasner, when asked to think of a name for a 1
followed by 100 zeros. 10^100 is an incredibly large number. The
largest reasonable estimates for the number of particles in the
universe is only about 10^85. A googol is a million times a billion
times this much.
I've been asked so many times that I finally tracked down the
answer to "Whatever happened to Milton Sirotta?" From his obituary,
it seems he died in 1980, I also found a note that said, "Edwin(his
brother) and Milton worked for most of their lives in their
father's factory in Brooklyn, NY, pulverizing apricot pits into an
abrasive used for industrial purposes".
Hectare A unit of land equal to 100 ares or about 2.47 acres. An
are is the area of a square with sides of a dekameter (ten meters).
A hectare is equal to a square made up of a ten by ten array of
ares or, in more modern words, a square with sides of 100 meters.
The prefix hecto is from the Greek word for one hundred, hekaton.
The prefix is common in units of measure, such as hectogram or
hectometer.
Helix is preserved from the Greek and has maintained its meaning
since antiquity. The Greek word seems to have been used generally
to apply to ideas about wrapping or twisting, but only its
mathematical meaning seems to have survived.
Histogram The root of histogram is probably from the Greek root
histo, for tissue, and gram, for write or draw. The suffixes gram
and graph are almost interchangeable, and both have to do with the
act of writing or drawing. Karl Pearson, the first known user,
apparently thought of each vertical bar as a cell. Some have
suggested that the root is from the word "history" since a
histogram provides a record, and certainly Pearson knew of this
meaning also.
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In a recent post to a math history news group, John Aldrich of
the Department of Economics of the University of Southampton
wrote:
I do not know the history of the technique but the _term_
histogram was coined by Karl Pearson to refer to a "common form of
graphical representation". The _OED_ quotes from _Philos. Trans. R.
Soc._ A. CLXXXVI, (1895) 399 [Note. The word "histogram" was]
introduced by the writer in his lectures on statistics as a term
for a common form of graphical representation, i.e., by columns
marking as areas the frequency corresponding to the range of their
base. Stigler (_History of Statistics_) identifies the lectures as
the 1892 lectures on the geometry of statistics. The Greek root of
history is from histor, a learned man. The implication is that a
learned man is aware of history, but it is more direct than just
good advice. The Indo-European root of the word is the same root
that gives us wise.
Hour and year are both derived from the Greek root horo, which
was applied to ideas about time and the seasons. In the Old
Germanic horo became yero and year was thus derived from the same
root which gave us hour. Today horoscope refers to fortune telling,
but the practice is rooted in the original meaning, measuring the
aspect of the stars and planets to measure the seasons. Horology is
still the name for a maker of timepieces.
Hundred is from the German root hundt. The quantity that it
represents has not been consistent over the years and has ranged
from its present value, 100, to 112, 120, 124, and 132 at different
times in different areas. The remnants of these old measures still
persist in the hundredweight of some countries representing 112 or
120 pounds, depending on the country. A hundred has also been used
to represent an area of land equal to 100 hides (of cattle?). The
measure of area was frequently used in colonial US, and parts of
England in place of "Shire" or "Ward". A curious custom related to
one hundred as a unit of land occurs in England when a member of
the House of Commons wishes to resign his seat, which is illegal.
An MP accepts stewardship of the
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"Chiltern Hundreds", an area of chalk hills near Oxford and
Buckingham, and effects his release from Parliament.
Hypotenuse comes from the common Greek root hypo(for under, as
in hypodermic -under the skin) and the less common tein or ten, for
stretch. This last is the source of our modern word tension. The
hypotenuse was the line segment "stretched under" the right
angle.
The other two sides of a right triangle are generally called
legs, but the term is also applied to any side of a triangle with
the idea that they are standing upon a "base". The term, leg, is
also applied to either of the two parts on each side of the vertex
of a curve such as a parabola or hyperbola. More formally, I have
recently seen the term cathetus used to describe the two
non-hypotenuse sides of a right triangle. Cathetus actually means a
straight line falling perpendicularly on another straight line or
surface, and was used by Euclid in this fashion in his tenth
definition in the first book of The Elements
When a straight line standing on a straight line makes the
adjacent angles equal to one another, each of the equal angles is
right, and the straight line standing on the other is called a
perpendicular to that on which it stands.
The literal meaning of the Greek root is "to let down". The
medical term cathetor, and the electrical term cathode both come
from the same root.
At Jeff Miller's website I found :
Cathetus occurs in English in 1571 in A Geometricall Practise
named Pantometria by Thomas Digges (1546?-1595) (although it is
spelled Kathetus). Cathetus is found in English in the Appendix to
the 1618 edition of Edward Wright's translation of Napier's
Descriptio. The writer of the Appendix is anonymous, but may have
been Oughtred.
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[[My thanks to Steve Earth, math teacher at the Kehillah
Jewish High School, for suggesting this term]]
Isosceles is the union of the Greek iso (same or equal) and
skelos (legs) and refers to two sides of a object as being the same
length, as in isosceles triangles and isosceles trapezoids. The
root iso shows up in many scientific and mathematical words such as
isometry (same measure), and isomorphic (same shape). Isobar is
used both in chemistry (two atoms with equal atomic weight) and in
meteorology
(lines connecting points of equal barometric pressure). The two
equal length sides are called legs (see above), and the other side
is called the base.
I recently (2009) became aware that the term "arms" is sometimes
used instead of legs. Here is a link to a
January The first month of the year was originally a period of
festival between the end of one year and the beginning of the next
in honor of the Roman god Janus. Janus was the god of
beginnings and endings and is portrayed with two faces, one
looking forward and one looking back.
Logarithm is the combination of two Greek roots, Logos(reason or
ratio) + artihmus(number). The ratio refers to the original method
of constructing logarithms by geometric sequences. The name was
introduced by John Napier (1550-1617), the inventor of logarithms,
in his 1614 work on logarithms, Mirifici logarithmorum canonis
descriptio, [Description of the wonderful canon of logarithms ....
but it is usually called "The Descripto"]. It was originally
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written in Latin and subsequently translated into English. Here
is a site where you can find a digital copy of the English text
It seems that Pietro Mengoli (1625-1686) was the first to use
the term "natural logarithm". Boyer writes, "Mercator took over
from Mengoli the name 'natural logarithm' for values that are
derived by means of this series." The term Mengoli and Mercator
actually used was "logarithmus naturalis". In a discussion group,
Jeff Miller suggested that it might be this use of noun before
adjective that prompted the use of the symbol "ln" for natural log
rather than "nl". According to Cajori, the symbol "ln" was first
used for the natural logarithm (log base e) in 1893 by Irving
Stringham (1847-1909). Stringham introduces the notation without
comment in a list of symbols following the table of contents, then
uses it for the first time on page 41, shown below.
Thanks to Dave Renfro for help in getting this digital pic.
I also recently heard in a correspondence from George Zeliger
that when he was a student in Russia (around 1989) it was common to
use "lg" for the common logarithm (log base ten).
When Napier constructed his tables he used a base that was
slightly smaller than one (1-10-7) and so as the number, n, got
bigger, the logarithm, l, got smaller. It was common at the time in
trigonometry tables to divide the radius of a circle into
10,000,000 parts. Because the main intention of his creation was
focused on addressing the difficulty in performing trigonometric
computations, Napier also divided his basic unit into 107 parts.
Then to avoid having to use fractions, he multiplied each value by
107. In notation of today's mathematics, the form of Napier's logs
would look like : 107 (1-10-7)L=N. Then L is the Naperian logarithm
of N.
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According to e: The Story of a Number by Eli Maor,
In the second edition of Edward Wright's translation of Napier's
Descripto (London, 1618), in an appendix probably written by
William Oughtred, there appears the equivalent of the statement
that loge10 = 2.302585. Since the actual tables contains no
decimals it was probably given as 2302585 without the decimal
point.
In a famous meeting between Napier and Henry Briggs, Briggs
suggested the use of a base of 10 instead of 1- 10-7 and to have
the logarithm of one equal to zero. This Napier agreed to but the
task of constructing tables of "common" logarithms fell to Briggs,
and they were often called Brigg's Logarithms in his honor.
Robin Wilson, in his Gresham College lecture on the number e,
that "Early ideas of logarithms are given in works of Chuquet and
Stifel around the year 1500. They listed the first few powers of 2
and noticed that to multiply any two of them it is enough to add
their exponents." Maor notes that Joost Burgi of Switzerland
probably created a table of logarithms before Napier by several
years, but did not publish until later, and he is almost forgotten
today. Burgi may also have independently discovered the method of
Prosthaphaeresis and gave it to Tycho Brahe. Burgi is also
remembered as the person who taught Kepler Algebra.
The impact of logartihms on the working scientist of the period
is hard to appreciate, but one may get an idea from this quote by
Pierre Laplace, "Logarithms, by shortening the labors, doubled the
life of the astronomer." While it is Napier's work on logartihms
that he is remembered for today, in his own time he was famous for
the calculating method called Napiers rods and a method of
calculating spherical right triangle trigonometry. He thought his
most important work had been published 21 years earlier in 1593. In
that year he published a mathematical analysis of the book of
Revelations in the Bible, A Plaine Discovery of the Whole
Revelation of Saint John. In the book he revealed that the Pope was
the antichrist, and that the world would end in the year 1786.
Fortunately for us, he was wrong on at least that one point. To his
credit, he more accurately predicted the development of the machine
gun, the submarine, and the tank.
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Gordon Fisher recently posted a time line of the development of
the use of the abbreviation "log" for lograrithms. Here is his post
with a few notes thrown in Log. (with a period, capital "L") was
used by Johannes Kepler (1571-1630) in 1624 in Chilias
logarithmorum (Cajori vol. 2, page 105) log. (with a period, lower
case "l") was used by Bonaventura Cavalieri (1598-1647) in
Directorium generale Vranometricum in 1632 (Cajori vol. 2, page
106). log (without a period, lower case "l") appears in the 1647
edition of Clavis mathematicae by William Oughtred (1574-1660)
(Cajori vol. 1, page 193). Kline (page 378) says Leibniz introduced
the notation log x (showing no period), but he does not give a
source. loga was introduced by Edmund Gunter (1581-1626) according
to an Internet source. [I do not see a reference for this in
Cajori.] Many students (and teachers) have heard colorful
legends
about the reasoning behind the use of "ln" for the natural
logarithm (from the French for something, or something about
the name Napier). Most of them seem to me to be more myth
than fact. The facts, as best I know them, is that the first
use of the terms "natural" and "logarithm" together was by
Nicholas Mercator (not the cartographer) in 1668 in his
logarithmo technica in which he used the Latinized "log
naturalis". [[[In early 2005 a post from Jeff Miller pointed
out that, according to Carl Boyer, Pietro Mengoli used the
term before Mercator. Both were working with values derived
from a series, Mercator with the expansion of log(1+x)]]]
The first use of "ln" as a symbol was, as Gordon points
out(below), by Stringham (I have not seen this book and do
not know if he gives an explanation). As to the correct
pronunciation of "ln(x)", whatever your teacher says is
correct, but high school students should be aware that many
college mathematicians find the symbol disturbing. In his
1984 biography, Paul Halmos described the symbol as
"childish". It is, however, very commonly used in computer
science. ln (for natural logarithm) was used in 1893 by Irving
Stringham (1847-1909) in Uniplanar Algebra (Cajori vol. 2, page
107). The same note from Jeff Miller mentioned above pointed
out
that Anton Steinhauser used the abbreviation "log.nat." in
1875 William Oughtred (1574-1660) used a minus sign over the
characteristic of a logarithm in the Clavis Mathematicae (Key to
Mathematics), "except in the 1631 edition which does
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not consider logarithms" (Cajori vol. 2, page 110). The Clavis
Mathematicae was composed around 1628 and published in 1631 (Smith
1958, page 393). Cajori shows a use from the 1652 edition.
I also recently saw a post that suggested that in computer
classes it is sometimes common to use "lg" for the base two
log.
In 1647 the French mathematician Saint-Vincent showed that the
area under the hyperbola y = 1/x were like the logarithm function,
that is, the area from 1 to 2 plus the area from 1 to 3 was equal
to the area from 1 to 6, 2x3.
Minute When the early sailors from the Eastern Mediterranean
chose to cut an arc into parts, they chose fractions in the
sexagesimal (base 60) system that was common to their period in
history. A nice article and illustrations of the Babylonian system
of numerals is found at the St. Andrews Math-History page. Later
when Latin writers described these small parts of an arc, they used
the Latin phrase pars minuta, Latin for small parts. Our unit of
time for 1/60 of an hour adopted and contracted this phrase into
minute. The Conjugate word with the same spelling but different
accent and pronunciation (mi nyoot') continues to refer to
something very small. The word MINUS for subtraction is drawn from
the same root and refers to making something smaller. 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.
Other related words are minor (smaller of two), minced (cut into
small pieces), miniature (on a small scale) and menu (a small
list).
Multiply comes from the combined roots of multi, many, and pli,
for folds, as in a number folded on itself many times. The first
use I have found of the word as a verb, as in "multiply two by
three" is credited to Chaucer in his 1391 work, A Treatise on the
Astrolabe.
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The two numbers that are multiplied together are most often
called factors and the result is called the Product. Although they
are not used much anymore, you may still find the two parts that
are multiplied together called the multiplicand [that which is
multiplied, or how many in each group] and the multiplier[that
which does the multipling, or how many groups in all].
One of the earliest notations to indicate multiplication was by
juxtaposition, placing the numbers adjacent to each other as we do
for algebraic characters today. Cajori cites this as the method
used to indicate multiplication on some ancient Indian manuscripts
from the 10th century or earlier. Jeff Miller has a note that "In
1553, Michael Stifel brought out a revised edition of Rudolff's
Coss, in which he showed multiplication by juxtaposition and
repeating a letter to designate powers (Cajori vol. 1, pages
145-147)."
The use of an "x" to indicate the operation of multiplication
seems to have been originated by William Oughtred in his Clavis
Mathematicae (Key to Mathematics, 1631). The use of a dot, as in 6
.4 = 24, is sometimes credited to Leibniz with the first use
attributed to a letter from Leibniz to John Bernoulli :
The dot was introduced as a symbol for multiplication by G. W.
Leibniz. On July 29, 1698, he wrote in a letter to John Bernoulli:
"I do not like X as a symbol for multiplication, as it is easily
confounded with x; ... often I simply relate two quantities by an
interposed dot and indicate multiplication by ZC LM. Hence, in
designating ratio I use not one point but two points, which I use
at the same time for division." [A History of Mathematical
Notation, Vol 1, art. 233; F. Cajori] From Jeff Millers web page on
"Earliest Uses of Symbols of Operation" I found the following
correction to Cajori; "Cajori shows the symbol as a raised dot.
However, according to Margherita Barile, consulting Gerhardt's
edition of Leibniz's Mathematische Schriften (G. Olms, 1971), the
dot is never raised, but is located at the bottom of the line. She
writes that the non-raised dot as a symbol for multiplication
appears in all the letters of 1698, and earlier, and, according to
the same edition, it already appears in a letter by Johann
Bernoulli to Leibniz dated September, 2nd 1694 (see vol. III, part
1, page 148). Some people credit the first use of a dot for
multiplication to Thomas Harriot. He used a dot in Analytica Praxis
ad
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Aequationes Algebraicas Resolvendas, which was published
posthumously in 1631. Cajori suggests these were not acutally
intended as symbols for the operation of multiplication but "Scott
(page 128) writes that Harriot was 'in the habit of using the dot
to denote multiplication.' And Eves (page 231) writes, 'Although
Harriot on occasion used the dot for multiplication, this symbol
was not prominently used until Leibniz adopted it." [from Jeff
Miller's page]. The use of a * instead of a dot appeared in
Teutsche Algebra (1659) by Johann Rahnn.
Some notes on notation for multiplicaton, The ancient Greeks and
Egyptians seemed to have no special symbol for multiplication.
Sometimes a word or phrase was used as we might say "times" to
indicate multiply. In the 16th Century Stifel used the capital M
and D for multiply and divide in his Deutsche Arithmetica (1548).
Other German writers did not follow his lead, and it seems that
Stifel quickly dropped the symbols himself. Simon Stevin adopted
the M and D in L' arithmetique(1634). Cajori credits the use by
Christian Wolf and Euler in the 18th Century with making the dot
popular in Europe, and the strong influence of Oughtred led to the
more common use of the "x" in England, and in America. In America
today it seems that "x" is more common through the teaching of
arithmetic, and the dot is introduced for awhile in the early
algebra teaching; but eventually the use of juxtaposition of
variables, and parentheses for numbers becomes the most common
indiation of multiplication. 3 x 4 = 3; 3 . 4; 3 (4)
Negative
Negative numbers, and the equivalent word for negative were
introduce by Brahmagupta, a Hindu mathematician around 600 AD. The
Latin root of today's word is negare, to deny. The negative
numbers, in this sense, denying or invalidating an equivalent
positive quantity.
The negative numbers were themselves denied for a long part of
mathematical history, and only slowly came to be accepted. The
first record of the operational rules for what we today call
positive and negative numbers came from the pen of Diophantus
(around 250 AD) who referred to them as "forthcomings" and
"wantings". His work may have been drawn from proposition five in
Euclid's Book II of the Elements in
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which Euclid demonstrates with geometric figures what we would
write in modern algebra as (a+b)(a-b)+b2 = a2. This, of course, is
easily recast as the more common identity (a+b)(a-b)= a2 - b2.
Diophantus would accept negatives only as a way of diminishing a
greater quantity, but did not accept them as independent quantities
and would not accept a solution that was negative. Al-Khwarizmi
(850 AD), whose writings brought Arabic numerals to the west, used
a similar approach with negatives allowed in-process but not as a
final result.
Descartes, around 1636, used the French fausse, false, for
negative solutions. Thomas Harriot had described negative roots as
the solution to an alternate form of the equation with the signs of
the odd powers changed. Today his idea would be expressed by saying
that the appearance of -c as a root of f(x) was only to be
understood to mean that c is a root of f(-x).
In Mathematics: The Loss of Certainty, by Morris Kline includes
the following argument against negative numbers by Antoine Arnauld
(1612-1694), mathematician, theologian, and friend of Blaise
Pascal; "Arnauld questioned that -1:1 = 1:-1 because, he said, -1
is less than +1; hence, how could a smaller be to a greater as a
greater is to a smaller?"
Franz Lemmermeyer wrote in a posting to the Historia-Matematica
newsgroup that Gleanings from the History of the Negative Number by
PGJ Vrendenduin suggests that a number line with both positive and
negative numbers could be found in the work of Wallis (1657)[This
is certainly true as seen here]. Another posting to the same list
quoted Kline's "Mathematical Thought from Ancient to Modern
Times":
"Though Wallis was advanced for his times and accepted negative
numbers, he thought they were larger than infinity but not less
than zero. In his 'Arithmetica Infinitorum' (1665), he argued that
since the ratio a/0, when a is positive, is infinite, then, when
the denominator is changed to a negative number, as in a/b with b
negative, the ratio must be greater than infinity." Even as late as
1831, De Morgan would still write that one "must recollect that the
signs + and - are not quantities, but directions to add and
subtract." [ Albrecht Heeffer refutes this position, held by Kline
and many others, in a
post to the math-history list. ] In a recent book by Gert
Schubring see clips here he also supports a view that
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Wallis' understanding of negatives was much broader than
generally credited.
According to a post from Laura Laurencich, the Incas had a
method of indicating both positive and negative numbers on their
quipus as documented by the Jesuit Priest Blas Valeria in 1618