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18TH CENTURY
MATHEMATICS - BERNOULLI BROTHERS
Unusually
in the history of mathematics, a single family, the Bernoulli’s, produced half
a dozen outstanding mathematicians over a couple of generations at the end of
the 17th and start of the 18th Century.
The
Bernoulli family was a prosperous family of traders and scholars from the free
city of Basel in Switzerland, which at that time was the great commercial hub
of central Europe.
The
brothers, Jacob and Johann Bernoulli, however, flouted their father's wishes
for them to take over the family spice business or to enter respectable
professions like medicine or the ministry, and began studying mathematics
together.
After
Johann graduated from Basel University, the two developed a rather jealous and
competitive relationship.
Johann
in particular was jealous of the elder Jacob's position as professor at Basel
University, and the two often attempted to outdo each other.
After
Jacob's early death from tuberculosis, Johann took over his brother's position,
one of his young students being the great Swiss mathematician Leonhard Euler.
However,
Johann merely shifted his jealousy toward his own talented son, Daniel (at one
point, Johann published a book based on Daniel's work, even changing the date
to make it look as though his book had been published before his son's).
Johann
received a taste of his own medicine, though, when his student Guillaume de
l'Hôpital published a book in his own name consisting almost entirely of
Johann's lectures, including his now famous rule about 0 ÷ 0 (a problem which
had dogged mathematicians since Brahmagupta's initial work on the rules for
dealing with zero back in the 7th Century).
This
showed that 0 ÷ 0 does not equal zero, does not equal 1, does not equal
infinity, and is not even undefined, but is "indeterminate" (meaning
it could equal any number).
The
rule is still usually known as l'Hôpital's Rule, and not Bernoulli's Rule.
Despite
their competitive and combative personal relationship, though, the brothers
both had a clear aptitude for mathematics at a high level, and constantly
challenged and inspired each other.
They
established an early correspondence with Gottfried Leibniz, and were among the
first mathematicians to not only study and understand infinitesimal calculus
but to apply it to various problems.
They
became instrumental in disseminating the newly-discovered knowledge of
calculus, and helping to make it the cornerstone of mathematics it has become
today.
But
they were more than just disciples of Leibniz, and they also made their own
important contributions.
One
well known and topical problem of the day to which they applied themselves was
that of designing a sloping ramp which would allow a ball to roll from the top
to the bottom in the fastest possible time.
Johann
Bernoulli demonstrated through calculus that neither a straight ramp or a
curved ramp with a very steep initial slope were optimal, but actually a less
steep curved ramp known as a brachistochrone curve (a kind of upside-down
cycloid, similar to the path followed by a point on a moving bicycle wheel) is
the curve of fastest descent.
This
application was an example of the “calculus of variations,” a generalization of
infinitesimal calculus that the Bernoulli brothers developed together, and has
since proved useful in fields as diverse as engineering, financial investment,
architecture and construction, and even space travel.
Johann
also derived the equation for a catenary curve, such as that formed by a chain
hanging between two posts, a problem presented to him by his brother Jacob.
Jacob
Bernoulli’s book “The Art of Conjecture”, published posthumously in 1713,
consolidated existing knowledge on probability theory and expected values, as
well as adding personal contributions, such as his theory of permutations and
combinations, Bernoulli trials and Bernoulli distribution, and some important
elements of number theory, such as the Bernoulli Numbers sequence.
He
also published papers on transcendental curves, and became the first person to
develop the technique for solving separable differential equations (the set of
non-linear, but solvable, differential equations are now named after him).
He
invented polar coordinates (a method of describing the location of points in
space using angles and distances) and was the first to use the word “integral”
to refer to the area under a curve.
Jacob
Bernoulli also discovered the appropximate value of the irrational number e while exploring the compound interest on loans.
When
compounded at 100% interest annually, $1.00 becomes $2.00 after one year; when
compounded semi-annually it ppoduces $2.25; compounded quarterly $2.44; monthly
$2.61; weekly $2.69; daily $2.71; etc.
If
it were to be compounded continuously, the $1.00 would tend towards a value of
$2.7182818... after a year, a value which became known as e.
Alegbraically,
it is the value of the infinite series (1 + 1⁄1)1.(1
+ 1⁄2)2.(1 + 1⁄3)3.(1
+ 1⁄4)4...
Johann’s
sons Nicolaus, Daniel and Johann II, and even his grandchildren Jacob II and
Johann III, were all accomplished mathematicians and teachers.
Daniel
Bernoulli, in particular, is well known for his work on fluid mechanics
(especially Bernoulli’s Principle on the inverse relationship between the speed
and pressure of a fluid or gas), as much as for his work on probability and
statistics.
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