Although the first set of rules for dealing with negative
numbers was stated in the 7th century by the Indian mathematician
Brahmagupta, it is surprising that in 1758 the British
mathematician Francis Maseres was claiming that negative
numbers
"... darken the very whole
doctrines of the equations and make dark of the things which are in
their nature excessively obvious and simple" .
Maseres and his contemporary, William Friend took the view
that negative numbers did not exist. However, other mathematicians
around the same time had decided that negative numbers could be
used as long as they had been eliminated during the calculations
where they appeared.


It was not until the 19th century when British mathematicians like
De Morgan, Peacock, and others, began to investigate the 'laws of
arithmetic' in terms of logical definitions that the problem of
negative numbers was finally sorted out.
However, there were references to negative numbers far
earlier...

In 200 BCE the Chinese number rod system (see note1 below)
represented positive numbers in Red and Negative numbers in black.
An article describing this system can be found here . These were
used for commercial and tax calculations where the black cancelled
out the red. The amount sold was positive (because of receiving
money) and the amount spent in purchasing something was negative
(because of paying out); so a money balance was positive, and a
deficit negative. 
The concept also appeared in Astronomy where the ideas of
'strong' and 'weak' were used for approximating a number from above
or below. For example approaching 5 from above means for example,
starting with 5.2 you can find better approximations 5.1, 5.05,
5.025. Thus 5.025 was called a 'strong' approximation and a number
like 4.9 'weak'. So 'strong' numbers were called positive and
'weak' numbers negative
In
India , negative numbers
did not appear until about 620 CE in the work of Brahmagupta (598 
670) who used the ideas of 'fortunes' and 'debts' for positive and
negative. By this time a system based on placevalue was
established in India, with zero being used in the Indian number
sytem. Brahmagupta used a special sign for negatives and stated the
rules for dealing with positive and negative quantities as
follows:
A debt minus
zero is a debt.
A fortune minus
zero is a fortune.
Zero minus zero
is a zero.
A debt
subtracted from zero is a fortune.
A fortune
subtracted from zero is a debt.
The product of
zero multiplied by a debt or fortune is zero.
The product of
zero multiplied by zero is zero.
The product or
quotient of two fortunes is one fortune.
The product or
quotient of two debts is one fortune.
The product or
quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a
debt is a debt.


The conflict between geometry and algebra
The ancient Greeks did
not really address the problem of negative numbers, because their
mathematics was founded on geometrical ideas. Lengths, areas, and
volumes resulting from geometrical constructions necessarily all
had to be positive. Their proofs consisted of logical arguments
based on the idea of magnitude. Magnitudes were represented by a
line or an area, and not by a number (like 4.3 metres or 26.5 cubic
centimetres). In this way they could deal with 'awkward' numbers
like square roots by representing them as a line. For example, you
can draw the diagonal of a square without having to measure it (see
note 2 below).
About 300 CE, the Alexandrian mathematician Diophantus (200  c.284
CE) wrote his
Arithmetica
, a collection of problems where he developed a series of symbols
to represent the 'unknown' in a problem, and powers of numbers. He
dealt with what we now call linear and quadratic equations. In one
problem Diophantus wrote the equivalent of 4 = 4x + 20 which would
give a negative result, and he called this result 'absurd'.
In the 9th century in
Baghdad Al  Khwarizmi (c.780  c.850
CE) presented six standard forms for linear or quadratic equations
and produced solutions using algebraic methods and geometrical
diagrams. In his algebraic methodshe acknowledged that he derived
ideas from the work of Brahmagupta and therefore was happy with the
notion of negative numbers. However, his geometrical models (based
on the work of Greek mathematicians) persuaded him that negative
results were meaningless (how can you have a negative square?). In
a separate treatise on the laws of inheritance, AlKhwarizmi
represents negative quantities as debts.
In the 10th century Abul Wafa (940998 CE) used negative numbers
to represent a debt in his work on 'what is necessary from the
science of arithmetic for scribes and businessmen'?. This seems to
be the only place where negative numbers have been found in
medieval Arabic mathematics. AbulWafa gives a general rule and
gives a special case where subtraction of 5 from 3 gives a "debt"
of 2. He then multiples this by 10 to obtain a "debt" of 20, which
when added to a 'fortune' of 35 gives 15.
In the 12th century Al  Samawal (1130  1180) had produced an
algebra where he stated that:
 if we subtract a positive number from an 'empty power', the
same negative number remains,
 if we subtract the negative number from an 'empty power', the
same positive number remains,
 the product of a negative number by a positive number is
negative, and by a negative number is positive.
Negative numbers did not begin to appear in Europe until the
15th century when scholars began to study and translate the ancient
texts that had been recovered from Islamic and Byzantine sources.
This began a process of building on ideas that had gone before, and
the major spur to the development in mathematics was the problem of
solving quadratic and cubic equations.
As we have seen, practical applications of mathematics often
motivate new ideas and the negative number concept was kept alive
as a useful device by the Franciscan friar Luca Pacioli (1445 
1517) in his Summa
published in 1494, where he is credited with inventing double entry
bookkeeping.
Solving equations
The story of the solution of
equations begins in Italy in the 16th century (see note 3 below).
This story is full of intrigue and deception because methods of
solution were kept secret. The issue which caused most
consternation at the time was the meaning of $\sqrt{1}$. In fact,
Cardano (1501  1576) in his Ars
Magna of 1545 had to solve a problem where $\sqrt{15}$
appeared.
Cardano found a sensible answer (see note 4 below) by working
through the algorithm, but he called these numbers 'ficticious'
because not only did they disappear during the calculation, but
they did not seem to have any real meaning. However, by 1572, the
Italian engineer, Bombelli (1526  1572) had provided the correct
rules for working with these 'imaginary' numbers(see note 5
below).
In the 17th and 18th century, while they might not have been
comfortable with their 'meaning' many mathematicians were routinely
working with negative and imaginary numbers in the theory of
equations and in the development of the calculus.
The English mathematician, John Wallis (1616  1703) is credited
with giving some meaning to negative numbers by inventing the
number line, and in the early 18th century a controversy ensued
between Leibniz, Johan Bernoulli, Euler and d'Alembert about
whether $\log (x)$ was the same as Log(x).
By the beginning of the 19th century Caspar Wessel (1745  1818)
and Jean Argand (1768  1822) had produced different mathematical
representations of 'imaginary'numbers, and around the same time
Augustus De Morgan (1806  1871), George Peacock (1791  1858)
William Hamilton (1805  1865) and others began to work on the
'logic'of arithmetic and algebra and a clearer definition of
negative numbers, imaginary quantities, and the nature of the
operations on them began to emerge.
Negative numbers and imaginaries are now built into the
mathematical models of the physical world of science, engineering
and the commercial world.
There are many applications of negative numbers today in
banking, commodity markets, electrical engineering, and anywhere we
use a frame of reference as in coordinate geometry, or relativity
theory.
Pedagogical Note:
It seems that the problems that people had (and now have  see the
Lottery incident ) in understanding the use of negative numbers
concerns:
 the difference between the operation of subtraction and the
object (a negative
number), since the same sign is used for both
 the language involved like 'minus minus 3' as opposed to
'subtract negative 3'
 separating the physical model or analogy (be it profit/loss or
rise/fall in temperature or rotation/direction in the plane) from
the rules of operating on the entities.
References
English
Mathematicians
Francis Maseres (1731  1824)
A dissertation on the use of the negative sign in algebra.
(1758)
Fellow of Clare College Cambridge and Fellow of the Royal
Society
William Frend
Principles of Algebra (1796)
Printed by J. Davis, for G. G. and J. Robinson, Paternoster
Row
Other
Sources
Berggen, J.L. (1986)
Episodes in the Mathematics of
Mediaeval Islam . SpringerVerlag N.Y. andBerlin.
Menninger, K. (1969)
Number Words and Number
Symbols . M.I.T. Press Cambridge, Mass. andLondon.
Schubring, G. (2005)
Conflicts Between Generalization,
Rigor, and Intuition: Number Concepts Underlying the Development of
Analysis in 17  19th Century France and Germany .
SpringerVerlag N.Y.
Ifrah, G. (1998)
The
Universal History of Numbers . Harvill Press, London.
Li Yan and Du Shiran (Tr. Crossley, J.N and Lun A. W. ) (1987)
Chinese Mathematics: a
Concise History . O.U.P. Oxford.
Web
References
Mactutor at St Andrews University
http://wwwhistory.mcs.stand.ac.uk/history/
.
Notes
 The counting rod system was certainly in operation in the
period (475  221 BCE)  called the period of the 'Warring States'
[Yan andShiran 1987, 7/8])
 In our notation, $\sqrt{2}$ and $\sqrt{5}$ occurred when
finding the diagonal of a square or constructing the Golden
Section.
 The period from Pacioli (1494) to Descartes (1637), a period of
about 150 years brings the solution of equations to a stage where
they could be understood by school pupils today.
 In modern notation, Cardano's multiplication was $(5\sqrt{15}
)(5+ \sqrt{15} )$, and applying the rule for brackets this becomes
$25  15 = 40$.
 Even though mathematicians did not find a suitable
representation for negative numbers, it did not prevent them from
following the ordinary rules of arithmetic and developing rules for
the imaginary numbers as well. This is where the beauty of
mathematical invention is not limited by the 'real' world.