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Introduction
On the topic of mathematical symbols.....
"Every meaningful mathematical statement can also be
expressed in plain language. Many plain-language statements of
mathematical expressions would fill several pages, while to
express them in mathematical notation might take as little as one
line. One of the ways to achieve this remarkable compression is
to use symbols to stand for statements, instructions and so
on."
Lancelot Hogben
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The symbol n!,
called factorial n, was introduced in 1808 by Christian Kramp of
Strassbourg, who chose it so as to circumvent printing
difficulties incurred by the previously used symbol thus
illustrated on the right. (Eves p132)
The symbol n! for "factorial n", now universally used
in algebra, is due to Christian Kramp (1760-1826) of Strassburg,
who used it in 1808. (Cajori p341)
EVES, HOWARD "Great Moments in Mathematics - Before
1650", Mathematical Association of America 1983.
CAJORI, FLORIAN "A History of Mathematics", The
Macmillan Company 1926.
Our familiar signs, in
geometry, for similar (on the left), and for congruent (on the
right) are due to Leibniz (1646-1715.) (Eves p253)
Leibniz made important contributions to the notation of
mathematics. In Leibnizian manuscripts occurs this symbol (on the
left) for “similar,” and this symbol (on the right) for
“equal and similar” or “congruent.” (Cajori
p211)
EVES, HOWARD "An introduction to the History of
Mathematics," fourth edition, Holt Rinehart Winston 1976
CAJORI,FLORIAN "A History of Mathematics", The
Macmillan Company 1926
In 1923, the National Committee
on Mathematical Requirements, sponsored by the Mathematical
Association of America, recommended this symbol (on the left) as
standard usage for angle in the United States. Historically,
Pierre Herigone, in a French work in 1634, was apparently the
first person to use a symbol for angle. He used both the symbol
above as well as this symbol on the right, which had already been
used to mean "less than." The standard symbol survived,
along with other variants, as follows.
These appeared in
England circa 1750.
During the 19th
century in Europe these forms were used to designate the angle
ABC, and the angle between a and b , respectively.
This symbol, representing the arc on the
angle, first appeared in Germany in the latter half of the 19th
century.
This symbol (on
the left) for right angle was used as early as 1698 by Samuel
Reyher, who symbolized "angle B is a right angle" as
illustrated on the right, using the vertical line for equality.
This commonly used symbol for right angle
appeared in America around 1880 in the widely used Wentworth
geometry textbook. (NCTM p362,364)
THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical
Topics for the Mathematics Classroom", National Council of
Teachers of Mathematics (USA) 1969
This symbol for pi was used by the early English
mathematicians William Oughtred (1574 -1660), Isaac Barrow
(1630-1677), and David Gregory (1661-1701) to designate the
circumference , or periphery, of a circle. The first to use the
symbol for the ratio of the circumference to the diameter was the
English writer, William Jones, in a publication in 1706. The
symbol was not generally used in this sense, however, until Euler
(1707-1783) adopted it in 1737. (Eves p99)
Oughtred's notation was the forerunner of the relation pi =
3.14159..., first used by William Jones in 1706 in his Synopsis
palmariorum matheseos. Euler first used pi = 3.14159... in 1737.
In his time, the symbol met with general adoption. (Cajori p158)
This symbol for pi was used by Oughtred in an expression to
represent the ratio of the diameter to the circumference. Isaac
Barrow, from 1664, used the same symbolism. David Gregory used pi
in an expression to represent the ratio of the circumference to
the radius in 1697. The first to use pi definitely to stand for
the ratio of circumference to diameter was an English writer
William Jones. He used it to symbolize the word
"periphery." Euler adopted the symbol in 1737, and
since that time it has been in general use. (Smith p312)
The number pi is the ratio of the circumference of a circle to
its diameter. It is also the ratio of the area of a circle to the
area of the square on its radius. The adoption of the symbol for
pi for this ratio is essentially due to the usage given it by
Leonhard Euler from 1736 on. In the 1730's, Euler first used p
and c for the circumference -to-diameter ratio, then adopted this
symbol for pi. However, he is not the originator of the symbol.
An actual ratio symbol as illustrated here
on the right had been used by William Oughtred in 1647 and by
Isaac Barrow in 1664 to indicate the ratio of the diameter of a
circle to it's circumference or periphery.
David Gregory, nephew of Scottish
mathematician James Gregory (1638-1675), used this symbol on the
left for the ratio of circumference to radius in 1697. In 1706
the English writer William Jones, in a work that gave the
100-place approximation of John Machin, first used the single
symbol for pi. This computation of pi to a large number of places
by means of various series representations was aided by the use
of such relations as pi/4 = 4 arctan (1/5) - arctan (1/239), as
given by Machin in 1706. (NCTM p148,152)
EVES, HOWARD "An Introduction to the History of
Mathematics," fourth edition, Holt Rinehart Winston 1976.
CAJORI, FLORIAN "A History of Mathematics", The
Macmillan Company 1926
SMITH, D.E. "History of Mathematics" volume II. Dover
Publications 1958
THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical
Topics for the Mathematics Classroom". National Council of
Teachers of Mathematics (USA) 1969
Percent has been used since the end of the fifteenth
century in business problems such as computing interest, profit
and loss, and taxes. However, the idea had its origin much
earlier. When the Roman emperor Augustus levied a tax on all
goods sold at auction, centesima rerum venalium, the rate was
1/100. Other Roman taxes were 1/20 on every freed slave and 1/25
on every slave sold. Without recognising percentages as such,
they used fractions easily reduced to hundredths.
In the Middle Ages, as large
denominations of money came to be used, 100 became a common base
for computation. Italian manuscripts of the fifteenth century
contained such expressions as "20 p 100" and "x p
cento" to indicate 20 percent and 10 percent. When
commercial arithmetics appeared near the end of that century, use
of percent was well estasblished. For example, Giorgio Chiarino
(1481) used "xx. per .c." for 20 percent and "viii
in x perceto" for 8 to 10 percent. During the sixteenth and
seventeenth century, percent was used freely for computing profit
and loss and interest. (NCTM p146,147}
In its primitive form the per
cent sign is found in the 15th century manuscripts on commercial
arithmetic, where it appears as this symbol after the word
"per" or after the letter "p" as a
contraction for "per cento." The use of the per cent
symbol can be seen in this extract from an anonymous Italian
manuscript of 1684 (Smith p250)
The percent sign, %, has probably evolved
from a symbol introduced in an anonymous Italian manuscript of
1425. Instead of "per 100," "P cento," which
were common at that time, this author used the symbol shown.
By about 1650, part of this symbol had
been changed to the form shown on the right. Finally, the
"per" was dropped, leaving this symbol to stand alone,
and this in turn became %. (NCTM p147)
The solidus form (%) is modern. (Smith p250)
This symbol stands for "per
thousand". (Hogben p92)
It is natural to expect that percentage
will develop into per millage, and indeed this has not only
begun, but it has historic sanction. Bonds are quoted in New York
using this symbol on the right, and so in other commercial lines.
At present, indeed, the symbol above (Hogben) is used in certain
parts of the world, notably by German merchants, to mean
"per mill," a curious analogue to % developed without
regard to the historic meaning of the latter symbol.(Smith p250)
THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical
Topics for the Mathematics Classroom", National Council of
Teachers of Mathematics (USA) 1969
SMITH, D.E. "History of Mathematics" volume II, Dover
Publications 1958
HOGBEN, LANCELOT "The Wonderful World of Mathematics",
Macdonald and Company 1968
The Anglo-American symbol for division is of 17th
century origin, and has long been used on the continent of Europe
to indicate subtraction. Like most elementary combinations of
lines and points, the symbol is old. It was used as early as the
10th century for the word est. When written after the letter
"i", it symbolized "id est." When written
after the word "it", it symbolized
"interest." 
If written after
the word "divisa", for "divisa est", this
might possibly have suggested its use as a symbol of division.
Towards the close of the 15th century the Lombard merchants used
it to indicate a half, along with similar expressions such as
this one on the right.
There is also a possibility that it was
used by some Italian algebrists to indicate division. In a
manuscript entitled Arithmetica and Practtica by Giacomo Filippo
Biodi dal Aucisco, copied in 1684, this symbol stands for
division, suggesting that various forms of this kind were
probably used.
The Anglo-American symbol (above top) first appeared in print in
the Teutsche Algebra by Johann Heinrich Rahn (1622-1676) which
appeared in Zurich in 1659. This symbol was then made known in
England by the translation of Rahn's work by Dr. John Pell in
London in 1688. (Smith p406)
Around the year 1200, both the Arabic writer al-Hassar, and
Fibonacci (Leonardo of Pisa), symbolised division in fraction
form with the use of a horizontal bar, but it is thought likely
that Fibonacci adopted al-Hassar's introduction of this
symbolisation.
In his Arithmetica integra (1544) Michael
Stifel employed the arrangement 8)24 to mean 24 divided by 8.
(NCTM p139)
Michael Stifel (1486?-1567) was regarded as the greatest German
algebrist of the 16th century. (Cajori p140)
SMITH, D.E. "History of Mathematics" volume II, Dover
Publications 1958
THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical
Topics for the Mathematics Classroom", National Council of
Teachers of Mathematics (USA) 1969
CAJORI, FLORIAN "A History of Mathematics", The
Macmillan Company 1926
Thomas Harriot (1560-1621) was
an English mathematician who lived the longer part of his life in
the sixteenth century but whose outstanding publication appeared
in the seventeenth century. He is of special interest to
Americans, because in 1585 he was sent by Sir Walter Raleigh to
the new world to survey and map what was then Virginia but is now
North Carolina. As a mathematician Harriot is usually considered
the founder of the English school of algebraists. His great work
in this field, the Artis Analyticae Praxis was published in
London posthumously in 1631, and deals largely with the theory of
equations. In it he makes use of these symbols above,
">" for "is greater than" (on the left),
and "<" for "is less than" (on the right.)
They were not
immediately accepted, for many writers preferred these symbols,
which another Englishman William Oughtred (1574-1660) had
suggested in the same year in the popular Clavis Mathematicae, a
work on arithmetic and algebra that did much toward spreading
mathematical knowledge in that country.
Isaac Barrow (1630-1677), in a book Lectiones Opticae &
Geometricae (London 1674), used these symbols as follows: 
this meant "A major est quam B"
and this meant "A minor est quam
B."
These symbols to the right are modern and are not
international.
The symbol on the left means "is not equal to."
The middle symbol means "is not less than."
The symbol on the right stands for "is not greater
than."
In the 1647
edition of Oughtred's Clavis mathematicae these somewhat
analogous symbols appear for "non majus" (on the left)
and "non minus" (on the right) respectively.
On the Continent
these symbols, or some of their variants, apparently invented in
1734 by the French geodesist Pierre Bouguer (1698-1758), are
commonly used. Bouguer was one of the French geodesists sent to
Peru to measure an arc of a meridian. (Eves p251, Smith p413)
EVES, HOWARD "An introduction to the History of
Mathematics," fourth edition, Holt Rinehart Winston 1976
SMITH, D.E. "History of Mathematics" volume II, Dover
Publications 1958
John Wallis (1616-1703) was one of the most original
English mathematicians of his day. He was educated for the Church
at Cambridge and entered Holy Orders, but his genius was employed
chiefly in the study of mathematics. The Arithmetica infinitorum,
published in 1655, is his greatest work. (Cajori p183)
This symbol for infinity is first found in print in his 1655
publication Arithmetica Infinitorum. It may have been suggested
by the fact that the Romans commonly used this symbol for a
thousand, just as today the word “myriad” is used for
any large number, although in the Greek it meant ten thousand.
The symbol was used in expressions such as, in 1695, "jam
numerus incrementorum est (infinity)." (Smith p413)
The symbol for infinity, first chosen by John Wallis in 1655,
stands for a concept which has given mathematicians problems
since the time of the ancient Greeks. A case in point is that of
Zeno of Elea (in southern Italy) who, in the 5th century BC,
proposed four paradoxes which addressed whether magnitudes
(lengths or numbers) are infinitely divisible or made up of a
large number of small indivisible parts. (Brinkworth and Scott
p80)
Wallis thought of a triangle, base
length B, as composed of an infinite number of “very
thin” parallelograms whose areas (from vertex to base of the
triangle) form an arithmetic progression with 0 for the first
term and ( A /(infinity)). B for the last term - since the last
parallelogram (along the base B of the triangle) has altitude
(A/(infinity)) and base B.
The area of the triangle is the sum of the arithmetic progression
O + . . . . + (A/(infinity)).B
= (number of terms/2). (first + last term)
=(infinity/2).(0+(A/(infinity)).B)
=(infinity/2).(A/(infinity)).B
=(A-B)/2
(NCTM p413)
CAJORI, FLORIAN "A History of Mathematics", The
Macmillan Company 1926
SMITH, D.E. "History of Mathematics" volume II, Dover
Publications 1958
BRINKWORTH & SCOTT "The Making of Mathematics", The
Australian Association of Mathematics Inc. 1994
THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical
Topics for the Mathematics Classroom", National Council of
Teachers of Mathematics (USA) 1969
The symbol : to indicate ratio seems to
have originated in England early in the 17th century. It appears
in a text entitled Johnson’s Arithmetick ; In two Bookes
(London.1633), but to indicate a fraction, three quarters being
written 3:4. To indicate a ratio it appears in an astronomical
work, the Harmonicon Coeleste (London, 1651), by Vincent Wing. In
this work the forms A : B :: C : D and A.B :: C.D appear
frequently as being equal in meaning. (Smith p406)
William Oughtred (1547-1660)
was another English mathematician who wrote as follows:
A : B = C : D as A B :: C D.
He laid extraordinary emphasis upon the use of mathematical
symbols; altogether he used over 150 of them. Only 3 have come
down to modern times, and one of these is this symbol for
proportion. His notation for ratio and proportion was later
widely used in England and on the Continent. (Cajori p157).
In his Clavis Mathematicae (1631) Oughtred used the dot to
indicate either division or ratio, but in his Canones Sinuum
(1657) the colon : is used for ratio. He wrote 62496 : 34295 :: 1
: 0 / 54.9- (Smith p 407)
As this notation gained ground it freed the dot . for use as the
symbol for separation in decimal fractions. It is interesting to
note the attitude of Leibniz (1646-1715) toward some of these
symbols. On July 29, 1698, he wrote in a letter to John Bernoulli
thus ".... in designating ratio I use not one point but two
points, which I use at the same time, for division; thus for your
dy.x :: dt.a I write dy:x = dt:a; for dy is to x as dt is to a,
is indeed the same as, dy divided by x is equal to dt divided by
a. From this equation follow then all the rules of
proportion.” This conception of ratio and proportion was far
in advance of that in contemporary arithmetics. (Cajori p158)
It is possible that Leibniz, who used : as a general symbol for
division, took it from these writers, for he wrote in 1684
“x : y quod idem est ac x divis. Per y seu x/y.”
The hypothesis that the ratio symbol : came from the symbol for
division by dropping the bar has no historical basis. Since it is
more international than the division symbol, it is probable that
the latter symbol will gradually disappear. Various other symbols
have been used to indicate division, but they have no particular
interest at the present time. (Smith p407)
Ratio - the quotient of two numbers or quantities indicating
their relative sizes. The ratio of a to b is written a : b or
a/b. The first term is the antecedent and the second the
consequent. (Daintith and Nelson p274)
The symbol :: for the equality of ratios, now giving way to the
common sign for equality, was introduced by Oughtred circa 1628,
for he later wrote "proportio, sive ratio aequalis ::"
and a Dr. Pell gave it still more standing when he issued Rahn's
algebra in English in 1668. The symbol seems to have been
arbirarily chosen.
This symbol for continued proportion was used by English writers
of the 17th and 18th centuries. For example it was used by Isaac
Barrow (1630-1677) in his Lectiones Mathematicae (London, 1683),
where he wrote "The character is made use of to signify
continued Proportionals." It is still commonly seen in
French textbooks. (Smith p413)
SMITH, D.E. "History of Mathematics" volume II, Dover
Publications 1958
CAJORI, FLORIAN "A History of Mathematics", The
Macmillan Company 1926
DAINTITH, JOHN and NELSON,R.D. "Dictionary of
Mathematics", Penguin 1989
Although the great practical invention of zero has
often been attributed to the Hindus, partial or limited
developments of the zero concept are clearly evident in a variety
of other numeration systems that are at least as early as the
Hindu system, if not earlier. The actual effect of any one of
these earlier steps in the full development of the zero concept -
or, indeed, whether there was any actual effect - is by no means
clear, however.
The Babylonian sexagesimal system used in the mathematical and
astronomical texts was essentially a positional system, even
though the zero concept was not fully developed. Many of the
Babylonian tablets indicate only a space between groups of
symbols if a particular power of sixty was not needed, so the
exact powers of sixty that were involved must be determined
partly by context. In the later Babylonian tablets (those of the
last three centuries B.C.) a symbol was used to indicate a
missing power, but this was used only inside a numerical grouping
and not at the end. (NCTMp49)
Not to be overlooked is the fact that in
the sexagesimal notation of integers the "principle of
position" was employed. Thus, in 1.4 (=64), the 1 is made to
stand for 60, the unit of the second order, by virtue of its
position with respect to the 4. The introduction of this
principle at so early a date is the more remarkable, because in
the decimal notation it was not regurlarly introduced until about
the ninth century after Christ. The principle of position, in its
general and systemic application, requires a symbol for zero. We
ask, Did the Babylonians possess one? Had they already taken the
gigantic step of representing by a symbol the absence of units?
Babylonian records of many centuries later -of about 200
B.C.-give a symbol for zero which denoted the absence of a
figure, but apparently it was not used in calculation. It
consisted of two angular marks as illustrated above on the right,
one above the other, roughly resembling two dots, hastily
written. About 130 A.D. Ptolemy in Alexandria used in his
Almagest the Babylonian sexagesimal fractions, and also the
omicron o to represent blanks in the sexagesimal numbers. This o
was not used as a regular zero. It appears therefore that the
Babylonians had the principle of local value, and also a symbol
for zero, to indicate the absence of a figure, but did not use
this zero in computation.Their sexagesimal fractions were
introduced into India and with these fractions probably passed
the principle of local value and the restricted use of the zero.
(Cajori p5)
When the Greeks
continued the development of astronomical tables, they explicitly
chose the Babylonian sexagesimal system to express their
fractions, rather than the unit-fraction system of the Egyptians.
The repeated subdivision of a part into 60 smaller parts
necessitated that sometimes “no parts” of a given unit
were involved, so Ptolemy’s tables in the Almagest (c. A.D.
150) included both of these symbols for such a designation.
Considerably later, in approximately 500,
Greek texts used this symbol, the omicron, the first letter of
the Greek word ouden (“nothing”). Earlier usage would
have restricted the omicron to symbolizing 70, its value in the
regular alphabetic arrangement.
Perhaps the earliest systematic use of a
symbol for zero in a place-value system is found in the
mathematics of the Mayas of Central and South America. The Mayan
zero symbol was used to indicate the absence of any units of the
various orders of the modified base-twenty system. This system
was probably used much more for recording calendar times than for
computational purposes. (NCTM p49)
The Maya counted essentially on a scale of 20, using for their
basal numerals two elements, a dot representing one and a
horizontal dash representing five. The most important feature of
their system was their zero, this character as illustrated, which
also had numerous variants. (Smith p44)
It is possible that the earliest Hindu
symbol for zero was the heavy dot that appears in the Bakhshali
manuscript, whose contents may date back to the third or fourth
century A.D., although some historians place it as late as the
twelfth. Any association of the more common small circle of the
Hindus with the symbol used by the Greeks would be only a matter
of conjecture. (NCTM p50)
There is no probability that
the origin will ever be known, and there is no particular reason
why it should be. We simply know that the world felt the need of
a better number system, and that the zero appeared in India as
early as the 9th century, and probably some time before that, and
was very likely a Hindu invention. In the various forms of
numerals used in India, and in later European and Oriental forms,
the zero is represented by a small circle or by a dot. Variations
include these, as illustrated. (Smith p70)
Since the earliest form of the Hindu symbol was commonly used in
inscriptions and manuscripts in order to mark a blank, it was
called sunya, meaning “void” or “empty.” This
word passed over into the Arabic as sifr, meaning
“vacant.” This was transliterated in about 1200 into
Latin with the sound but not the sense being kept, resulting in
zephirum or zephyrum. Various progressive changes of these forms,
including zeuero, zepiro, zero, cifra, and cifre, led to the
development of our words “zero” and “cipher.”
The double meaning of the word “cipher” today -
referring either to the zero symbol or to any of the digits - was
not in the original Hindu. In early English and American schools
the term “ciphering” referred to doing sums or other
computations in arithmetic. (NCTM p50)
The traditional Chinese numeration system is a base-ten system
employing nine numerals and additional symbols for the
place-value components of powers of ten. Before the eighth
century A.D. the place where a zero would be required was always
left absent. A circular symbol for zero is first found in a
document dating from 1247, but it may have been in use a hundred
years earlier. (NCTM p43)
Interestingly enough, the
forms of the modern Arabic numerals are not the same as the
Hindu-Arabic forms of the western world. For example, their
numerical representation for five is 0 and their zero is
representated by a dot. (NCTM p49)
This can be illustrated as shown; (Smith p70)
The various forms of the
numerals used in India after the zero appeared may be judged from
this table. (Smith p70)
This table illustrates some
later European and Oriental forms. (Smith p71)
The name for zero is not settled even yet. Older names and
variations include naught, tziphra, sipos, tsiphron, rota,
circulus, galgal, theca, null, and figura nihili.(Smith p71)
THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS, "Historical
Topics for the Mathematics Classroom", National Council of
Teachers of Mathematics (USA) 1969
CAJORI, FLORIAN "A History of Mathematics", The
Macmillan Company 1926
SMITH, D.E. "History of Mathematics" volume II, Dover
Publications 1958
The ancient writers commonly wrote the word for root
or side, as they wrote other words of similar kind when
mathematics was still in the rhetorical stage. The symbol most
commonly used by late medieval Latin writers to indicate a root
was R , a contraction of radix, and this, with numerous
variations, was continued in the printed books for more than a
century. Thus it appears as such in the works of Boncompagni
(1464), Chuquet (1484), Pacioli (1494), de la Roche (1520),
Cardan (1539), Tartaglia (1556), Ghaligai (1521), and Bombelli
(1572.) The symbol was also used for other purposes, including
response, res, ratio, rex and the familiar recipe in a
physician's prescription.
Meanwhile, the Arab writers had used
various symbols for expressing a root, including this sign on the
right, but none of them seem to have influenced European writers.
This symbol first appeared in print in
Rudolff's Coss in 1525, but without our modern indices. It is
frequently said that Rudolff used this sign because it resembled
a small "r", for radix (root), but there is no direct
evidence that this is true. The symbol may quite have been an
arbitrary invention. It is a fact , however, that in and after
the 14th century we find in manuscripts such forms as the
following for the letter "r."
It was a long time after these writers that a simple method was
developed for indicating any root, and then only as a result of
many experiments. French, English, and Italian writers of the
16th century were slow in accepting the German symbol, and indeed
the German writers themselves were not wholly favourable to it.
The letter l (for latus, side; that is, the side of a square) was
often used. In the 17th century our common square-root sign was
generally adopted, of course with many variants. The different
variants of the root sign are too numerous to mention in detail
in this work, particularly as they have little significance. By
the close of the 17th century the symbolism was, therefore,
becoming fairly well standardised, although there still remained
some work to be done. The 18th century saw this accomplished, and
it also saw the negative and fractional exponent come more
generally into use.
Some variations on the radical sign are as follows. The
illustrations are the work of many different writers, including
Stifel (1553), Gosselin (1557), Ramus-Schoner (1592), Rahn
(1659), Stevin (1585), Vlacq, Biondini (1689) and Newton (1707).
- for the square root
- for the cube root
-for the fourth root
-for the fifth root
-for the sixth root
-for the eighth root
(Smith p407 - p410)
SMITH, D.E. "History of Mathematics" volume II, Dover
Publications 1958
The symbols of elementary arithmetic are almost
wholly algebraic, most of them being transferred to the numerical
field only in the 19th century, partly to aid the printer in
setting up a page and partly because of the educational fashion
then dominant of demanding a written analysis for every problem.
When we study the genesis and development of the algebraic
symbols of operation, therefore, we include the study of the
symbols in arithmetic. Some idea of the status of the latter in
this respect may be obtained by looking at almost any of the
textbooks of the 17th and 18th centuries. Hodder in 1672 wrote
"note that a + (plus) sign doth signifie Addition, and two
lines thus = Equality, or Equation, but a X thus,
Multiplication," no other symbols being used. His was the
first English arithmetic to be reprinted in the American colonies
in Boston in 1710. Even Recorde (c1510-1558), who invented the
modern sign of equality, did not use it in his arithmetic, the
Ground of Artes (c1542), but only in his algebra, the Whetstone
of witte (1557). (Smith p395)
There is some
symbolism in Egyptian algebra. In the Rhind papyrus we find
symbols for plus and minus. The first of these symbols represents
a pair of legs walking from right to left, the normal direction
for Egyptian writing, and the other a pair of legs walking from
left to right, opposite to the direction for Egyptian writing.
[Eves 1, p42]
The earliest symbols of operation that have come down to us are
Egyptian. In the Ahmes Papyrus (c1550 B.C.) addition and
subtraction are indicated by these symbols on the left and right
above respectively.
The
Hindus at one time used a cross placed beside a number to
indicate a negative quantity, as in the Bakhshali manuscript of
possibly the 10th century. With this exception it was not until
the 12th century that they made use of the symbols of operation.
In the manuscripts of Bhaskara (c1150) a small circle or dot is
placed above a subtrahend as illustrated for -6, or the
subtrahend is enclosed in a circle to indicate 6 less than zero.
The early European symbols for plus are
listed opposite. The word plus, used in connection with addition
and with the Rule of False Position is not known before the
latter part of the 15th century.
The use
of the word minus as indicating an operation occurred much
earlier, as in the works of Fibonacci (c1175-1250) in1202. The
bar above the letter simply indicated an omission. In the 15th
century, this third symbol was also often used for minus, but
most writers preferred the other variations.
In the 16th
century the Latin races generally followed the Italian school,
using the letters p and m, each with the bar above it, or their
equivalents, for plus and minus. However, the German school
preferred these symbols, neither of which is found for this
purpose before the 15th century. In a manuscript of 1456, written
in Germany, the word "et" is used for addition and is
generally written so that it closely resembles the modern symbol
for addition. There seems little doubt that the sign is merely a
ligature for "et", much in the same way that we have
the ligature "&" for the word "and."
The origin of the
minus sign has been more of a subject of dispute. Some have
thought that it is a survival of the bar above the three symbols
for minus as listed above. It is more probably that it comes from
the habit of early scribes of using it as a shorthand equivalent
of "m." Thus Summa became Suma with the bar above the
letter u, and 10 thousand became an X with ther bar above the
letter. It is quite reasonable to think of the dash (-) as a
symbol for "m" (minus), just as the cross (+) is a
symbol for "et." Other forms of minus are here
illustrated.
There were other various written forms for plus and minus, as in
piu (Italian), mas (Spanish), plus (French) and et (German) for
plus and as in de or men (Italian), menos (Spanish), moins
(French) for minus. Examples of such usage include:
Pacioli (1494), Italian de or m for minus
Tartaglia (1556) and Catanes (1546), Italian, piu and men
Santa-Cruz (1594), Spanish, mas and menos
Peletier (1549), French, plus and moins
Gosselin (1577), P and M
Trenchant (1566), + and -
The expression "plus or
minus" is very old, having been in common use by the Romans
to indicate simply "more or less". It is often found on
Roman tombstones, where the age of the deceased is given as
illustrated to indicate "94 years, more or less".
These signs first
appeared in print in an arithmetic, but they were not employed as
symbols of operation. In the latter sense they appear in algebra
long before they do in arithmetic.They appeared in Johann
Widman's (c1460-?) arithmetic published in Leipzig in 1489, the
author saying: "Was - ist / das ist minus...vnd das + das
ist mer." He then speaks of "4 centner + 5 pfund,"
and also of "4 centner - 17 pfund," thus showing the
excess or deficiency in the weight of boxes or bales. (Smith p395
to 399)
Observe that Francis Vieta
(1540-1603) employed the Maltese cross (+) as the shorthand for
addition, and the (-) for subtraction. These two characters had
not been in very general use before his time. The introduction of
the + and - symbols seems to be due to the Germans, who, although
they did not enrich algebra during the Renaissance with great
inventions, as did the Italians, still cultivated it with great
zeal. The arithmetic of John Widmann, brought out in 1489 in
Leipzig, is the earliest printed book in which the + and -
symbols have been found, and the facsimile shown is from the
Augsburg edition of his work, dated 1526. The + sign is not
restricted by him to ordinary addition; it has the more general
meaning "et" or "and" as in the heading,
"regula augmenti + decrementi." The - sign is used to
indicate subtraction, but not regularly so. The word
"plus" does not occur in Widmann's text; the word
"minus" is used only two or three times. The symbols +
and - are used regularly for addition and subtraction, in 1521,
in the arithmetic of Grammateus, the work of Heinrich Schreiber,
a teacher at the University of Vienna. His pupil Christoff
Rudolff, the writer of the first text book on algebra in the
German language (printed in1525) employs these symbols. So did
Michael Stifel, who brought out an improved second edition of
Rudolff's book on algebra Die Coss in 1553. Thus, by slow
degrees, the adoption of the + and - symbols became universal.
Several independant paleograhic studies of Latin manuscripts of
the fourteenth and fifteenth centuries make it almost certain
that the + sign comes from the Latin et, as it was cursively
written in manuscripts just before the time of the invention of
printing. The origin of the sign - is still uncertain. (Cajori
p139)
The first one to make use of these signs
in writing an algebraic expression was the Dutch mathematician
Vander Hoecke, who in 1514 gave this illustration (on the left)
for radical three quarters minus radical three fifths, and for
radical 3 add 5 he gave the sign as shown on the right.
These symbols seem to have been employed
for the first time in arithmetic, to indicate operations, by
Georg Walckl in 1536. The illustration on the left indicates the
addition of one third of 230, and the one on the right indicates
the subtraction of one fifth of 460. From this time on the two
symbols were commonly used by both German and Dutch writers, the
particular signs themselves not being settled until well into the
18th century.
England adopted the Teutonic forms, and Robert Recorde
(c1510-1558) wrote (c1542) "thys fygure +, whiche betoketh
to muche, as this lyn, - plaine without a cross lyne, betokeneth
to lyttle". As symbols of operation most of the English
writers of this period reserved the + and - signs for algebra.
Thus Digges (1572) in his treatment of algebra: "Then shall
you ioyne them with this signe + Plus", and Hylles (1600)
says: "The badg or signe of addition is +," stating the
sum of 3 and 4 as "3 more 4 are 7," and writing 10___3
for "10 lesse 3." (Smith p399-402)
SMITH, D.E. "History of Mathematics" volume II, Dover
Publications 1958
EVES, HOWARD "An introduction to the History of
Mathematics," fourth edition, Holt Rinehart Winston 1976
CAJORI, FLORIAN "A History of Mathematics", The
Macmillan Company 1926
William
Oughtred (1574-1660) contributed vastly to the propagation of
mathematical knowledge in English by his treatises, the Clavis
Mathematicae, 1631, published in Latin (English edition 1647),
Circles of Proportion, 1632, and Trigonometrie, 1657. Among his
most noted pupils are the mathematician John Wallis (1616-1703)
and the astronomer Seth Ward.
Oughtred laid
extraordinary emphasis upon the use of mathematical symbols :
altogether he used over 150 of them. Only three have come down to
modern times, namely the cross symbol for multiplication, :: as
that of proportion, and the symbol for "difference
between." The cross symbol, on the left, occurs in the
Claris, but the letter X, seen on the right, which closely
resembles it, occurs as a sign of multiplication in the anonymous
"Appendix to the Logarithmes" in Edward Wright's
translation of John Napier's Descriptio, published in 1618. This
appendix was most probably written by Oughtred.
Leibniz (1646-1715) objected to the use of Oughtred's cross
symbol because of possible confusion with the letter X. On 29
July 1698 he wrote in a letter to John Bernoulli : "I do not
like (the cross) 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."
Through the aid of Christian Wolf (1679-1754) the dot was
generally adopted in the 18th century as a symbol for
multiplication. Wolf was a professor at Halle, and was ambitious
to figure as a successor of Leibniz. Presumably Leibniz had no
knowledge that Harriot in his Artis analyticae praxis, 1631, used
a dot for multiplication, as in aaa__3.bba=+2.ccc. Harriot's dot
received no attention, not even from Wallis. (Cajori p157)
The common symbol as illustrated was
developed in England about 1600. It was not a new sign, having
long been used in cross multiplication, in the check of nines,
where Hylles (1600) speaks of it as the "byas crosse"
in connection with the multiplication of terms in the division or
addition of fractions, for the purpose of indicating the
corresponding products in proportion, and in the "multiplica
in croce" of algebra as well as in arithmetic.
The symbol was not readily adopted by arithmeticians, being of no
practical value to them. In the 18th century some use was made of
it in numerical work, but it was not until the second half of the
19th century that it became popular in elementary arithmetic. On
account of its resemblance to x it was not well adapted to use in
algebra, and so the dot came to be employed, as in 2 . 3 = 6
(Europe) as well as in America. This device seems to have been
suggested by the old Florentine multiplication tables; at any
rate Adriaen Vlacq (c1600-1667), the Dutch computer (1628), used
it in some of his work, thus:
factores---- 7 . 17
faci---------119
although not as a real symbol of operation. In his text he uses a
rhetorical form, thus; "3041 per 10002 factus erit
30416082."
Christopher Clavius (1537-1612), a Jesuit
of Rome, wrote in 1583 using the idea of a dot for
multiplication, as in 3/5.4/7 for 3/5 X 4/7; and Thomas Harriot
(1560-1621) in a posthumous work of 1631 actually used the symbol
in a case like 2.aaa = 2a cubed. The first writer of prominence
to employ the dot in a general way for algebraic multiplication
seems to