Stephanie Meesdom
Stefanie Meul
Ellen Polfliet
Christian Huygens was born 14 April 1629 in The Hague, Netherlands. He was the second son of Constantin Huygens and Suzanna Van Baerle. It was through his father that Christian was to gain access to the top scientific circles of the times. In particular Constantin had many contacts in England and corresponded regularly with Mersenne and was a friend of Descartes.
Until he was sixteen, he was taught at home by private teachers. Descartes influenced his mathematical education because he was an occasional visitor at the Huygens’ home.
At the age of sixteen, in other words in 1645, he studied law and mathematics at the University of Leiden until 1647. From 1647 until 1649 he continued to study law and mathematics but now at the College of Orange at Breda. After he became familiar with the method of the Greeks, he was able to find solutions that Archimedes himself couldn’t find.
In 1649 he went to Denmark as a part of a diplomatic team He hoped to be able to go to Stockholm to visit Descartes but the weather did not allow him to make this journey.
In 1650 he completed the classic work about floating bodies. "De iis quae liquido supernatant" (about parts that stand out a liquid) but he didn’t publish it. In 1651 "Cyclometrae" showed the fallacy in methods proposed by Gregory of Saint Vincent, who had claimed to square the circle. He also published a way to calculate the area of a section out an ellipse or a hyperbola. In 1654 he completed the classic work about the circle. "De circuli magnitudine inventa" (about finding the circumference).
It was Huygens who wrote in 1652 as first a physical formula (a mathematical equation for a physical phenomenon). That’s why he can be considered as the founder of the mathematical physics.
Christian started at the mechanics that Galilei breath new life into with his research of fallen and thrown objects. By that mathematical strictness of his style of argumentation, he found in 1652 the right rules of collision and in 1659 the right formula for strength.
Huygens soon turned his attention to lens grinding and telescope construction. Around 1654 he devised a new and better way of grinding and polishing lenses.
March 215, 1655 Huygens detected, using one of his own lenses, the first moon of
Saturn : Titan. In this same year he made his first visit to Paris. He informed the mathematicians in Paris including Boulliau of his discovery. The following year he discovered the true shape of the rings of Saturn. By 1656 Huygens was able to confirm his ring theory to Boulliau and the results were reported to the Paris group. In Systema Saturnum (1659), Huygens explained the phases and changes in the shape of the ring. Also in 1656 he discovered the nebula of Orion. Huygens’ speculations about our surrounding universe are collected in his book "Cosmotheoros" (world viewer). But it has only been published after his death, in 1698. |
That Titan is Christian’s image, together with the ring, he said himself:
Ingenii vivent monumenta, inscriptaque
coelo |
They stay the signs of my ingenuity, and let
the names |
Having informed Pascal and Boulliau about his discovery of Titan in turn Huygens learnt of the work on probability carried out in a correspondence between Pascal and Fermat . On his return to Holland Huygens wrote a small work "De Ratiociniis in Ludo Aleae" (Van rekeningh in spelen van geluck) on the calculus of probabilities, the first printed work on the subject. It contains practical rules for the game of dice and as well it’s a contribution to the theory of probabilities. It counts also for his work on logarithms, intending to improve the tuning of musical instruments. His reputation was now so great that in 1665 Louis XIV offered him a pension if he would live in Paris, which accordingly then became his place of residence.
Around 1656 Christian invented the pendulum clock. It greatly increased the accuracy of time measurement. Part of this discoveries, he published in 1673 in his most important work : "Horologium oscillatorium sive de motu pendulorum". The first chapter is devoted to pendulum clocks. The second chapter contains a complete account of the descent of heavy bodies under their own weights in a vacuum, either vertically down or on smooth curves. The most renewing in this work, was the theory of the centrifugal force and the ingenious derivation of the law of collision. Huygens attempts for the first time in his work to study the dynamics of bodies rather than particles. |
In this work Huygens proves that the cycloid is tautochronous, an important theoretical result but one which had little practical application to the pendulum. In the third chapter he defines evolutes and involutes, proves some of their more elementary properties, and illustrates his methods by finding the evolutes of the cycloid and the parabola. These are the earliest instances in which the envelope of a moving line was determined.
The cycloid is the locus of a point on the rim of a circle rolling along a straight line. |
In 1660, Huygens returned to Paris and went to meetings of various scientific societies there. At these societies he met many mathematics including Roberval, Carcavi, Pascal, Pierre Petit, Desargues and Sorbière. When Pascal visitid him in December 1660, he showed him his telescopes.
In 1661 he wrote "Novus cyclus Harmonicus" (about the prohibition of consecutive fifths).
In 1661 he visited London. He continued his contacts with the newly forming Royal Society. He showed them his telescopes. In Londen, Christian saw Boyle’s vacuum pump and he carried out a number of Boyle’s experiments for himself. Huygens was elected to the Royal Society of Londen in 1663
In 1665 he learnt that the Royal Society was investigating other forms of clocks. Huygens started to experiment with clocks regulated by springs, but their accuracy was poorer than his pendulum clocks.
In 1666 Colbert invitated him to become part of the Académie Royale des Sciences where he stayed until 1680. He discovered that the Society was not organized yet. After meetings with Roberval, Carcavi, Frenicle de Bessy and Buot, the society moved to the Bibliothèque du Roi where Huygens took up residence. Huygens assumed leadership of the group basing much on his knowledge of the way the Royal Society operated in England.
In 1668 he sent to the Royal Society of London, in answer to a problem they had proposed, a memoir in which (simultaneously with Wallis and Wren) he proved by experiment that the momentum in a certain direction before the collision of two bodies is equal to the momentum in that direction after the collision. This was one of the points in mechanics on which Descartes had been mistaken.
From his youth Huygens’ health had never been robust and in 1670 he had a serious
illness which resulted him leaving Paris for Holland. Before he left Paris, believing
himself to be close to death he asked that his unpublished papers on mechanics should be
sent to the Royal Society. By 1671 Huygens returned to Paris. However in 1672 Louis XIV invaded the Low Countries and Huygens found himself in the extremely difficult position of being in an important position in Paris at a time France was at war with his own country. Scientists of this era felt themselves above political wars and Huygens was able, with much support from his friends, to continue his work. In 1672 Huygens and Leibniz met in Paris and thereafter Leibniz was a frequent visitor to the Académie. In this same year Huygens learnt of Newton’s work on the telescope and on light. He, quite wrongly, criticized Newton’s theory of light, in particular his theory of color. |
In 1673, Gottfried Wilhelm Leibniz, came to Huygens to learn mathematics.
Papin Denis worked as an assistant to Huygens around this time and after he left to work with Boyle, Huygens was joined by Tschirnhaus. In 1676 Huygens had to return to The Hague because he was ill again. In the two years he spent there, he studied the double refraction.
By 1678 Huygens returned to Paris. In that year he wrote his "Traité de la lumière" But it was only published in1690 followed by "Discours sur la cause de la pesanteur". In this work Huygens argued in favour of a wave theory of light. Huygens stated that an expending sphere of light behaves as if each point on the wave front were a new source of radiation of the same frequency and phase. However he became ill in 1679 and than again in 1681. Again after his health returned, he worked on a new marine clock during 1682.
In 1689, in England, Christian met Newton, Boyle and others in the Royal Society. Huygens had a great admiration for Newton but at the same time he did not believe the theory of universal gravitation which he said "…appears to me absurd" For Christian it was a step back in the darkness he had just combated.
Huygens took part in most of the controversies and challenges which then played so large a part in the mathematical world, and wrote several minor tracts. In one of these he investigated the form and properties of the catenary. In another he stated in general terms the rule for finding maxima and minima of which Fermat had made use.
All his demonstrations, like those of Newton, are rigidly geometrical, and he would seem to have made no use of the differential or fluxional calculus, though he admitted the validity of the methods used therein. Thus, even when first written, his works were expressed in an archaic language, and perhaps received less attention than their intrinsic merits deserved.
One has tried two times to erect a statue for Christian Huygens in The Hague. The first futile attempt, in 1868, had failed because they had not raised enough money. The second attempt, in 1905, lead to a draft, but the local council refused it.
Physics
The Pendulum Clock
Work in astronomy required accurate timekeeping, and this prompted Huygens to tackle this problem. In 1656 he patented the first pendulum clock, which greatly increased the accuracy of time measurement. The regular movement of a pendulum, was already researched by Galilei. When Galilei was old, he had suggested to use this regularity for operating a clock. But he had never built such a clock. Christiaan, who was very skillful and who had technical insight, invented a fork by which the pendulummovement could undisturbed operate in the gear wheels of a clock. |
This discovery made from the pendulum clock a success that pushed him in three directions. First, Huygens believed that a pendulum swinging in a large area would be more useful at sea to determine longitude and he invented the cyclical pendulum with this in mind. Second, he tried to make the period, in which the pendulum (a thread with a bullet) moved from left to right, independent of the result, by describing another pad to the bullet than a arc of a circle. Third, he tried to describe the pendulum better than a thread with a pointed masse at the end of it, that represented the bullet, because it wasn’t the reality.
Simple Harmonic Motion |
For small swings of a pendulum, the displacement x satisfies the equation of the Simple Harmonic Motion or SHM. The solution (which we might as well assume is zero at t = 0 ) is of the form x = Asin(wt) with A the constant amplitude and w the angular speed of the oscillation. Therefore the graph of x against t is a sine curve. We hereby ignore frictional resistance. |
If we plot x against the velocity x’ then we get a curve in what is called the Phase plane. For this case this curve is a closed curve and in fact is an ellipse. |
Christiaan Huygens had a lot to thank at the invention of the spring-regulated clock. Also the invention that the pad of the bullet is a cycloid helped him. Thanks to the theory of the pendulum centre, he suspected that the center of gravity of connected objects, is as low as possible. This thesis looked so self - evident to him that he also applied the thesis on colliding bodies. Before this, he had already invented the theory of colliding bodies. In the " Horolgium Oscillatorium sive de motu pendolorum " (1673) he described all this theory of pendulum motion.
The Theory of Colliding Bodies
The principle used to solve these collision problems was first enunciated by Huygens and involves the existence of something called the Coefficient of Restitution e.
Consider the problem of two balls with masses m_{1} and m_{2 }sliding
on a line with velocities u_{1} and u_{2} .
If they collide, then linear momentum is conserved. i.e. the velocities v_{1} and
v_{2} after collision satisfy :
m_{1} u_{1} + m_{2} u_{2} = m_{1 }v_{1}
+ m_{2} v_{2}
In addition the relative velocities before and after collision satisfy :
u_{1} - u_{2} = - e ( u_{1} - u_{2} )
The fact that e is independent of the size of the relative velocities and only
depends on the material of the balls is the underlying assumption in this model. It is at
least a reasonable first approximation of what happens in practice.
For balls moving in two or three dimensions, similar methods apply, except in these cases when a collision occurs between balls it is only the component of the relative velocity along the line of centers that is affected. The component perpendicular to the line of entrees remains unchanged. Similar considerations apply if the collision is with a " wall" or a "corner".
The Theory of Light
Christiaan Huygens also invented the mechanical theory of light, wherein the
transmission of light is considered as an expansion of waves in a delicate surrounding. This theory of waves is often indicated as the theory of undulation and is developed in the year1677. Simultaneously, Isaac Newton invented the emission or corpuscular theory. He assumed that a source of light transmits lots of small particles. A vibratish character of light is not mentioned in Huygens' theory at all ! His theory of light is based on a general principle: "Around every particle there has to arise a spherical wave, which this particle is the center." |
Christian also researched the reflection of light.
In secondary school, we learn in the physics lessons about reflection and there we learn
the lows of Huygens.
Act 1 : The invaded ray 1, the normal n, and the reflected ray 1', are situated
in the same plane. Act 2 : The angle of reflexion beta is equal to the angle of incidence alpha. Act 3 : The way followed by the light is reversible |
Together with his colleague Robert Hooke, Christian Huygens laid down the freezing point and the boiling point of water as reference points for the thermometer.
Astronomy
Huygens was interested in lensgrinding and telescope constructions. Around 1654 he devised a new and better way of grinding and polishing lenses. Using one of his own lenses, Huygens detected, in1655, the first moon of Saturn. It’s the largest moon of Saturn and it’s called "Titan". In the same year he made his first visit to Paris. He informed the mathematicians in Paris including Boulliau of his discovery and in turn Huygens learnt of the work on probability carried out in a correspondence between Pascal and Fermat.
He also discovered the true shape of the Rings of Saturn. However others had different theories including Roberval and Boulliau. Boulliau had failed to detect Saturn’s moon Titan so Huygens realised that he was using an inferior telescope. By 1656 Huygens was able to confirm his ring theory to Boulliau and the results were reported to the Paris group. In "Systema Saturnium" 1659, Huygens explained the phases and changes in the shape of the ring. Some, including the Jesuit Fabri, attacked not only Huygens theories but also his observations. However by 1665 even Fabri was persuaded to accept Huygens’ ring theory as improving telescopes confirmed his observation.
Huygens was the first one who saw visible details on the Mars area and his drawings of 18 nov. 1659 may be considered as the beginning of the Mars cartographic. His drawings made it possible to appoint the rotation time of Mars very specific.
In 1662 Huygens invented "The complexed ocular"
When Huygens was about 60 year old. He decided to publish his ideas about live on another planets. Huygens excused himself to the readers and he said that his book is only built on guesses. In 1695 his book "The Cosmotheoros" was ready, but Huygens would never experience the real publication. "The Cosmotheoros" was titled "Kosmothéros, sivé De Terris Coelestibus, eanumque estnatu, conjuctunae".
The Cosmotheoros is an ingenious, good readable and sounds sometimes very modern. Huygens got rid of the world image where in the earth stands still as the center of the universe. The book contains an argument about life on another planets. The reasoning is as following : based on the Copernicance world picture, we can establish that all the planets move in a circle motion around the sun. Besides the planets rotate around there axis, they receive the light from the sun and they are spherical. There is a lot of resemblance with the earth. Therefore the plants and the inhabitants of those planets should look alike the ones on the earth.
Music
Out of science, Christian Huygens took only time for music. Huygens proposed to divide the octave in 31 equal intervals.
Tones can be made by the vibration of strings. Such a vibration is never singular, it has also frequencies of higher natural tones. By passing them in an octave, they are divided. Their proportions become smaller than 2. In this way there are proportions for the five harmonies:
3:2 = 1,500 for the fifth
4:3 = 1,333 for the fourth
5:4 = 1,250 for the major third
6:5 = 1,200 for the minor third
5:3 = 1,667 for the sixth
There are holes between 1 and 1,200 for a second, and between 1,667 and 2 for a septum.
The question is which proportions should we take because there is room for discretion. In the seventeenth century that discretion became a real problem. The violists were able to choose the septum, but the harpsichordists weren’t. They were bound by a fixed, immovable tuning so that they couldn’t change the scale without sounding out of tune.
Christian found the series of thirty-one tones. He probably divided
the logarithms of 1,200 and 1,250, of 1,333 and 1,500 and of 1,600 and 1,667 by the
logarithm of 2,000 and then multiplied the quotients by numbers above twelve. He
chose thirty-one because the products would be close to a whole. |
We can see that we cannot bring all six harmonies at the same time close to a whole.
Only with twelve it works a bit: 3,16 close to 3 ; 3,84 close to 4 ; 4,98 close to 5 ;
7,02 close to 7 ; 8,13 close to 8 and
8,85 close to 9
Christian rejected the possibility of twelve because of his sharp sense of hearing, with which he could perceive little differences in pitch. He talked about the suffering of the ear. A series of 12 tones was too painful, but one of 31 was maybe better.
Sources
http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Huygens.html
http://www.in-berlin.de/User/jd/himmel/astro/Huygens-e.html
http://www.geocities.com/Athens/Acropolis/6681/huygens.htm
http://www.brown.edu/Students/OHJC/hm4/huygens.htm
http://www.clas.ufl.edu/users/rhatch/08sr-hygns-bio.htm
http://www.bsfcs.org/laurathomas/christiaan.htm
http://malvm1.mala.bc.ca/~mcneil/thuygens.htm
http://my.voyager.net/stargazer/huygens.html
http://math.truman.edu/~thammond/history/Huygens.html
http://es.rice.edu/ES/humsoc/Galileo/Catalog/Files/huygens.html
http://galileo.imss.firenze.it/museo/b/ehuygen.html
http://www.astro.virginia.edu/~eww6n/math/EulerianIntegraloftheSecondKind.html