history of mathematics - National STEM Centre

\ Mathematics was not one
iof Pierre Fermat's major
inactivities, in contrast to
jDescartes. Fermat (1601-
f1665) was a lawyer who ||
fworked in the Parliament:!
?of the city of Toulouse.
IFor him, mathematics was
8 An overview of La Geometrie
Fermat as well as Descartes did revolutionary work uniting geometry and algebra.
His work was of less influence than Descartes's because it was not published until
1679, after Fermat's death. His book Ad Locos Pianos et Solidos Isagoge, in which
for the first time a kind of introduction to analytical geometry can be found, was
probably written in 1629, that is, well before La Geometrie. In his book, he
discusses systematically simple equations with two unknown quantities and draws
the corresponding curves, lines, circles, ellipses, parabolas and hyperbolas.
Fermat goes much further than Descartes with the equations of curves. His
systematic approach to curves and equations is clearer, but this can be explained by
the fact that his intention with Isagoge was very different from Descartes's intention
with La Geometrie.
La Geometrie
• Finding solutions of geometrical
construction problems with the help
of algebra.
• The emphasis is on finding lengths.
Seen in an algebraic light this means that
the emphasis is on equations with one
unknown quantity.
Activity 8.11 Coordinate geometry
Ad Locos Pianos et Solidos Isagoge
• Studying curves with the help of
algebra.
• The emphasis is on the conic
sections of Apollonius.
Seen in an algebraic light this means
that the emphasis is on equations
with two unknown quantities.
In one respect, Fermat was clearly more traditional than Descartes; he applied
algebra as Viete meant algebra to be applied. This was the main reason that his
Isagoge was already considered old-fashioned when it was published.
Here is an example that shows how Fermat considered the hyperbola. He uses
vowels for the unknown quantities, A and E in this example, and consonants for
known quantities, Z in this example. Fermat shows the curves in Figure 8.3. Point I
is characterised by NP ( = A) and PI ( = E).
Fermat wrote: 'A.E = Z pi. The rectangles NPI and NMO are equal. Point I thus
describes a hyperbola with asymptotes NM and NR.' (Today we write xy = c
instead of A.E = Z pi.)
1 What is the meaning of the addition 'pi' in Z pi?
2 a Comment on this example. Notice that coordinate axes, quadrants, positive
and negative numbers are used and that the curves are complete.
b Compare this example with both Descartes's and today's mathematics.
Fermat wrote several mathematical works that were published after his death.
However, some of his work was widely known earlier, through his correspondence,
including a method for finding maxima and minima of curves which is so close to
the current method that Fermat is considered by many to be the founder of
differential calculus. His method for finding areas under specific curves was also far
ahead of his time. He was a brilliant amateur mathematician.
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