history of mathematics - National STEM Centre

Descartes
fViete (Vieta in Latin) was =
|a lawyer and worked at
jibe courts of the French
jlings Henry III and Henry
|V. His mathematical
jfaetivities mainly took
Ijplace in his spare time. In
PI is work entitled Isagoge
|[1591) he used letters for
line first time to represent
jljeneral quantities.
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Before Descartes, problems which involved comparing lengths, areas and volumes
were inconceivable. It was not the difficulty of solving the equations which was the
problem. People of that time could calculate the answers, if necessary to an accuracy
of two decimal places, even without modern calculation aids.
In the years before Descartes, the French mathematician Fran§ois Viete (1540-
1603) was very influential. In his work on algebra, he set a geometric requirement
for his variables. Freely interpreted, Viete's requirement amounted to the idea that
lengths, areas and volumes cannot be added. Lengths can only be added to lengths,
areas to areas and volumes to volumes. In short, you can only add, algebraically,
quantities of the same dimensions, called homogeneous quantities. The same holds
true for subtracting quantities from each other, but not for multiplying and dividing.
Viete's requirement is called the law of homogeneity.
Variables a and b are, from a geometrical standpoint, the lengths of line segments.
The algebraic expression a + b is equivalent to the line segment whose length is the
sum of the lengths of the line segments a and b. Similarly, a x b is the area of the
rectangle with sides a and b. The quantity a represents a cube. However, a + 3b
does not satisfy the law of homogeneity, because length is being added to volume.
Here is a translation of part of a chapter from Viete's writings.
On the law of homogeneous quantities and the comparison of quantities in
degrees and sorts. The first and general law with respect to equations or
relations, that, since it is based on homogeneous matters, is called the law
of the homogeneous quantities.
Homogeneous things can only be compared to homogeneous things. This
is because you have no idea how to compare heterogeneous things, as
Adrastussaid.
Therefore, if a quantity is added to another quantity, the result is
consequently homogeneous.
If a quantity is subtracted from another quantity, the result is consequently
homogeneous.
If a quantity is multiplied with another quantity, the result is consequently
heterogeneous with this and that quantity.
If a quantity is divided by another quantity, the result is consequently
heterogeneous.
In antiquity they didn't considerthis law and this caused the great
darkness and blindness of the old analysts.
Viete's work in bringing algebra and geometry together was a great step forward,
both in algebra and in geometry. Algebra gained a classical geometrical grounding.
Mathematicians could now formulate geometrical problems algebraically.
However, Viete's law of homogeneity forced algebra into a geometrical straitjacket.
Because of this law, expressions such as a 3 + 6a 2 +\2a had no geometric meaning.
Descartes's La Geometric helped mathematicians to escape from this straitjacket.