Daniel Bernoulli

(8 February 1700 -17 March 1782). A Swiss mathematician and a physicist. He was born in Groningen in the Netherlands. Son of Johann Bernoulli (one of the “early developers” of calculus) and a nephew of Jacob Bernoulli (who "was the first to discover the theory of probability"). Daniel Bernoulli published about 80 works, including 50 papers in the editions of the Petersburg Academy of Sciences and 10 prize-winning memoirs of the Paris Academy. But there was only one large treatise – his famous Hydrodynamica, which had a complicated and partially dramatic fate. He is particularly remembered for his applications of mathematics to mechanics especially fluid mechanics and pioneering work in probability and statics.

Johann Bernoulli tried to map out Daniel's life, selected a wife for him and decided he should be a merchant. His own father had tried a similar strategy but Johann had resisted - so did Daniel. However, Daniel spent considerable time with his father and learned much about the secrets of the Calculus which Johann had exploited to gain his fame. By the time Daniel was 13, Johann was reconciled to the fact that his son would never be a merchant but absolutely refused to allow him to take up mathematics as a profession as there was little or no money in it. He decreed that Daniel would become a doctor. For the next few years Daniel studied medicine but never gave up his mathematics. In time it became apparent that Daniel's interest in Mathematics was no passing fancy, so his father relented and tutored him. Among the many topics they talked about, one was to have a substantial influence on Daniel's future discoveries. It was called the "Law of Vis Viva Conservation" which today we know as the "Law of Conservation of Energy". The young Bernoulli found a kindred spirit in the English physician William Harvey who wrote in his book On the Movement of Heat and Blood in Animals that the heart was like a pump which forced blood to flow like a fluid through our arteries. Daniel was attracted to Harvey's work because it combined his two loves of mathematics and fluids whilst earning the medical degree his father expected of him.

After completing his medical studies at the age of 21, he sought an academic position so that he could further investigate the basic rules by which fluids move; something which had eluded his father and even the great Isaac Newton. (Johann Bernoulli never credited Newton with his discoveries in connection with the Calculus, instead giving the credit almost entirely to Gottfried Wilhelm Leibniz; another source of rivalry in the early eighteenth century.) Daniel applied for two chairs at Basel in anatomy and botany. These posts were awarded by lot, and unfortunately for Daniel, he lost out both times.

By the age of 23, Daniel was in Padua, Italy. Whilst recovering from illness he designed a ship's hour glass which would produce a reliable trickle of sand even in stormy weather. He submitted his design to the French Academy and took first prize. Meanwhile a friend, Christian Goldbach, arranged for some of Daniel's other work to be published under the title Some Mathematical Exercises.

When he was 25, Daniel returned home to Basel to find a letter from Empress Catherine I of Russia awaiting him, inviting him to become professor of mathematics at the Imperial Academy in St. Petersburg. At first Daniel was not keen to travel to such a distant land, but his elder brother Nikolas offered to go with him. Catherine was so keen to secure Daniel that she agreed to offer a second chair to Nikolas! Unfortunately, Nikolas died of tuberculosis a year later. At first Daniel thought of returning home but stayed when his father suggested that one of his own students, a certain Leonard Euler would make an able assistant for Daniel in his research. Bernoulli and Euler tried to discover more about the flow of fluids. They wanted to know about the relationship between the speed at which blood flows and its pressure. To investigate this, Daniel experimented by puncturing the wall of a pipe with a small open ended straw and noted that the height to which the fluid rose up the straw was related to fluid’s pressure in the pipe.

Soon physicians all over Europe were measuring patients blood pressure by sticking point-ended glass tubes directly into their arteries. It was not until about 170 years later, in 1896 that an Italian doctor discovered a less painful method which is still in use today. However, Bernoulli's method of measuring pressure is still used today in modern aircraft to measure the speed of the air passing the plane; that is its air speed.

The Fluid Equation

Taking his discoveries further, Daniel Bernoulli now returned to his earlier work on Conservation of Energy. It was known that a moving body exchanges its kinetic energy for potential energy when it gains height. Daniel realised that in a similar way, a moving fluid exchanges its kinetic energy for pressure. Mathematically this law is now written:

$\frac{1}{2} \rho u^2 + P = constant$

($\rho$ is the greek letter rho)

where $P$ is pressure, $\rho$ is the density of the fluid and $u$ is its velocity. A consequence of this law is that if the velocity increases then the pressure falls. This is exploited by the wing of an aeroplane which is designed to create an area above its surface where the air velocity increases. The pressure in this area is lower than that under the wing, so the wing is pushed upwards by the relatively higher pressure under the wing.

In 1738, Hydrodynamica was published. It is founded mainly on the principle of conservation of ‘living forces’ (that is, kinetic energy). Bernoulli preferred to use this principle not in its traditional form, received with hostility by Newtonians, but in Christiaan Huygens’s formulation that Bernoulli named the principle of equality between the actual descent and potential ascent: ‘If any number of weights begin to move in any way by the force of their own gravity, the velocities of the individual weights will be everywhere such that the products of the squares of these velocities multiplied by the appropriate masses, gathered together, are proportional to the vertical height through which the centre of gravity of the composite of the bodies descends, multiplied by the masses of all of them’.