Ang mga sumusunod na talata ay hango mula sa aklat na sinulat ni David Brewster noong 1841, na mababasa sa internet.
Galileo Galilei (1564-1642)
Galileo seems to have been desirous of following the profession of a painter: but his father had observed decided indications of early genius; and, though by no means able to afford it, he resolved to send him to the university to pursue the study of medicine. He accordingly enrolled himself as a scholar in arts at the university of Pisa, on the 5th of November, 1581,[5] and pursued his medical studies under the celebrated botanist Andrew Cæsalpinus, who filled the chair of medicine from 1567 to 1592.
Tycho Brahe (1546-1601)
His father, Otto Brahe, who was descended from a noble Swedish family, was in such straitened circumstances, that he resolved to educate his sons for the military profession; but Tycho seems to have disliked the choice that was made for him; and his next brother, Steno, who appears to have had a similar feeling, exchanged the sword for the more peaceful occupation of Privy Councillor to the King. The rest of his brothers, though of senatorial rank, do not seem to have extended the renown of their family; but their youngest sister, Sophia, is represented as an accomplished mathematician, and is said to have devoted her mind to astronomy as well as to the astrological reveries of the age.
Johannes Kepler (1571-1630)
John Kepler, the youngest of this illustrious band, was born at the imperial city of Weil, in[205] the duchy of Wirtemberg, on the 21st December 1571. His parents, Henry Kepler and Catherine Guldenmann, were both of noble family, but had been reduced to indigence by their own bad conduct. Henry Kepler had been long in the service of the Duke of Wirtemberg as a petty officer, and in that capacity had wasted his fortune. Upon setting out for the army, he left his wife in a state of pregnancy; and, at the end of seven months, she gave premature birth to John Kepler, who was, from this cause, a sickly child during the first years of his life. Being obliged to join the army in the Netherlands, his wife followed him into the field, and left her son, then five years old, under the charge of his grandfather at Limberg. Sometime afterwards he was attacked with the smallpox, and having with difficulty recovered from this severe malady, he was sent to school in 1577.
Sunday, July 08, 2007
Tuesday, February 14, 2006
Brain teasers and math problems
This page is lifted from Homepage of Padmanabhan.
This page has a variety of mathematics problems/brain teasers which I have come across and solved at different times. They have appeared in several publications
dating from antiquity and I have discussed some of them in a column called
"Playthemes" I used run for an Indian Science Magazine in the eighties.
Brain teasers buffs will know a fair fraction of them, but --- hopefully --- everyone will find something new. I have put them here because I like them. I plan to add more to this list periodically or even make subjectwise pages if/when I get time.
Quickies
This page has a variety of mathematics problems/brain teasers which I have come across and solved at different times. They have appeared in several publications
dating from antiquity and I have discussed some of them in a column called
"Playthemes" I used run for an Indian Science Magazine in the eighties.
Brain teasers buffs will know a fair fraction of them, but --- hopefully --- everyone will find something new. I have put them here because I like them. I plan to add more to this list periodically or even make subjectwise pages if/when I get time.
- Prove that the area of a cyclic quadrilateral of sides a,b,c,d is
A=sqrt[(s-a)(s-b)(s-c)(s-d)]. Of course, things become familiar when d goes to zero. - Take any n points in a plane and let D and d be the greatest and least distances determined by points of this set. Prove that 2D > sqrt(3) [sqrt(n)-1]d.
- A right angled tetrahedron ABCD is one in which the three angles at the vertex A are all right angles. Prove that the square of the area of the triangle BCD is the sums of the squares of the areas on the other three sides. [This can be done mentally in 10 seconds with the right approach.]
- If a cube is cut into finite number of smaller cubes, prove that at least two of them must be of same size.
- In a circumference of circle with center O and radius R, take three points EFG which are the vertices of an equilateral triangle. Draw three circles with centers E,F and G respectively with radius R. These three circles overlap pairwise forming three petals. Find the area of the overlapping region.
- Imagine two very long cylinders of unit radius located along the x-axis and y-axis respectively. Find the volume of the overlapping region. [No, it does not require calculus. Moreover the answer does not involve pi.]
- Let S be the set of all positive definite integers written in base 10. Remove from it all the integers which have the digit 9 appearing in them. Add up the reciprocals of the remaining ones. Is the sum finite or divergent? Remember that the sum of the reciprocals of all primes diverges.
- (a) A region of area A in a plane is bounded by a simply connected curve of length L. Is there a straightline passing through the region such that it divides both the area and perimeter into two equal parts?
(b) There are two nonoverlapping regions of arbitrary shapes, in a plane each bounded by simply connected curves. Is there a straight line passing through both and dividing both simultaneously bisecting the areas ? If "no", prove it; if "yes" give the construction. - A decagon ABC...IJ is inscribed in a circle. Show that the length of AD is the sum of the side of the decagon and the radius of the circle. (Sure, it is trivial to do this with trigonometry; but there is an elegant geometrical solution.)
- Let ABCD be a convex quadrilateral with perimeter P and the length of the longer diagonal L. What should be the shape of ABCD if (P/L) has the maximum possible value ? [No, it is not a square!]
- The diameter of a convex region of perimeter P is defined to be the length of the longest chord in that region. We can define the "pi" for any convex figure by taking the ratio of perimeter to diameter of that convex figure. Prove that this "pi" is bounded from below by 2 (trivial!) and above by actual pi.
- Compared to the last one this is surprisingly easy: Show that the area of any convex region of diameter D is bounded from above by (pi/4)D^2.
- A circular region of radius 16 cm has 650 points marked inside. You are given a circular ring of inner radius 2 cm and outer radius 3 cm. Show that for any choice of 650 points, you can always place the ring covering at least 10 points.
- What is the area of the largest ellipse that can be inscribed in a right angled triangle of sides a,b ? [Yes, there is an nice way of doing it].
- Given an acute angled triangle, inscribe in it a triangle such that it has the minimum possible perimeter. [There are elegant geometrical solutions; but if you are a physicist/engineer it should be trivial]
- These three are classics but I have come across people who haven't seen them. So ....
(a) In any triangle prove that the nine-point circle touches the incircle and the three excircles [If you don't know what a nine-point-circle is your education is incomplete].
(b) In triangle ABC the trisectors of the interior angles A,B,C are drawn making the adjacent ones meet at D,E,F [There is only one sensible way of doing this]. Prove that triangle DEF is equilateral.
(c) Start with an arbitrary triangle ABC. Draw three equilateral triangles ABX, BCY, CAZ on the outside on each side of the triangle. Show that the centers of these three equilateral triangles form an equilateral triangle. - In triangle ABC, AB=AC, angle A = 20 degrees, a point P is chosen along AC such that AP = BC. Find angle PBC.
- Construct a triangle given the three altitudes. [Before you start drawing a triangle with the 3 altitudes as sides, ask yourself: Will the three altitudes of a triangle always form a triangle ?]
- A point P is chosen inside a triangle ABC and the perpendiculars are dropped from P to the sides AC and BC.
(a) Prove that if x and y are the perpendicular distances then xy is a maximum when P lies on the bisector of angle C.
(b) Consider next the case when the perpendiculars are dropped from P to all the three sides with lengths x, y and z. Find P such that the product xyz is a maximum. [Sorry! It is not the incenter.] - Construct triangle ABC given the circumcenter, incenter and one excenter.
- When a beer can is full of beer its center of gravity is at the geometric center.When the beer can is empty it is again at the geometric center [assume the can is a perfect cylinder and any asymmetry between the top and bottom faces are ignorable]. As one starts to drink the beer, the CG, of course, starts to come down towards the bottom. It must reach a minimum height and rise again. If the beer weighs x grams and the empty can weighs y grams, find the lowest point reached by CG as a fraction of the height of the can.
- Two ladders of length a,b are leaning against opposite walls separated by distance x. Their feet rest at the opposite walls. If the height of intersection
of the ladders is d, find x in terms of (a, b, d). - Here is a politically incorrect cryptaithm: (EVE/DID)=.TALKTALKTALK..... Each letter stands for a digit and the right hand side is recurring as shown. Find the numbers.
- Show that the 13th of a month falls more frequently on Friday than on any other day.
- Let us assume you are a distance x units away from certain disaster in your life. Everything you do has a probability p of moving you 1 unit away from disaster and alas, a probability 1-p of taking you towards the disaster. What are your longterm chances of avoiding disaster in your life ? [Comment: Obviously, for p=0 you court disaster while if p=1 you will monotonically move away from it. Do you think p=0.3 persons are doing better than p=0.01 persons in their life ?]
- Every young man dating girls has faced this dilemma. "Should one move towards commitment or move away and hope that the next one will be better ?" Assume that: (a) You hope to consider a sample of N girls, one at a time. (b) You can't go back to someone you rejected. So if you pass up (N-1), you are stuck with the Nth for your life who could be a Medusa. At any given time, you have the rankings of all the previous girls you have dated to compare with the current one. What is your optimal strategy to get the best one ? [Ans: For reasonably large N, pass the first [N/e] however good they are and then choose the first one who is better than all those you have passed, without waiting further. If such a candidate doesn't come up, marry the Nth (Medusa) with the satisfaction that you did your best! I was never convinced "e" is natural until I solved this one!]
Quickies
- A point P is chosen inside a square ABCD such that angle PAB=angle PBA =15 degrees. Prove that PCD is equilateral.
- Rectangle ABCD has AB=3 BC. Points P,Q trisect AB and are ordered A-P-Q-B. Show that angle CAB+ angle CPB =angle CQB
- Give 1000 consecutive integers, none of which is a prime number.
- Prove that there exists numbers of the form n=p^q where (a) p and q are irrational and (b) n is rational.
- A book of 500 pages has 500 typos randomly distributed. What is the probability that page 29 has no typos ?
- (a) A point P is chosen inside an equilateral triangle such that its distances from the vertices are 3,4 and 5 centimeters. What is the size of the equilateral triangle? (b) For a non-quickie generalisation try the following: If a,b,c,d denote the three distances from the vertices and the side of the equilateral triangle show that 3(a^4+b^4+c^4+d^4)=(a^2+b^2+c^2+d^2)^2
- In a shuffled deck of cards what is the most probable position [from the top, say] for the first black ace ? What is the most probable position for the second black ace ?
- What is the rule behind ordering of these letters: z,x,c,v,b,n,m ?
- What about the set : o,t,t,f,f,s,s,e, ..... [Find the next few letters]
- Here are sets of numbers enclosed by square brackets. Fill the next set of numbers. [1],[1,1],[2,1],[1,2,1,1],[1,1,1,2,2,1],[3,1,2,2,1,1],
[.............] - Given any obtuse angled triangle is it possible to dissect it into smaller triangles, all of which are acute ? [The answer is "yes"]
- Express 64 using two 4's without the use of any mathematical symbols.
- Add a suitable mathematical symbol in the space between two and three in 2 3 so that the resulting expression has a value between two and three. [I know two ways of doing it]
- An old chestnut is to express numbers in terms of, say, 4 fours and mathematical symbols, like 97=4(4!)+(4/4); 71=(4!+4.4)/(.4). Give a general formula to express any positive integer in terms of four 4's and symbols.
- 1023 players (yes, not 1024) participate in a tournament in which each game produces a decisive winner. Players are eliminated by knock-out with byes being given when odd number of players occur at any given round. How many matches need to be played to find a winner ?
- One of the papers by the famous mathematician Littlewood published in a French journal concludes as follows (The sentences in the journal were, of course, in French; what I give below is the English translation).
1. I am greatly indebted to Prof. Risez for translating the present
paper.
2. I am greatly indebted to Prof. Risez for translating the previous foot note.
3. I am greatly indebted to Prof. Risez for translating the previous foot note.
Littlewood is completely ignorant of French language; so how did he avoid infinite regression?
Link
Friday, December 16, 2005
Judging Einstein
Before most physicists would believe the claims of relativity, they required proof—which would come in the form of a solar eclipse
J. Donald Fernie
This year we celebrate the centenary of Albert Einstein's special theory of relativity. Indeed, 1905 was the year in which Einstein first gave notice of his astonishing abilities. He was but 26 and had just earned his doctorate, but that year he published four papers on separate topics, each of which marked a major advance in physics. The first of these, on the photo-electric effect (the subject of Roald Hoffmann's Marginalium in the previous issue), would bring him the Nobel Prize, but it was the third, on special relativity, that made him both famous and controversial. A decade after this flurry of papers, in 1915, he unveiled the theory of general relativity, shaking again the foundations of science.
To test predictions of Einstein's theories... So different was relativity from the prevailing beliefs that most physicists demanded proof that it could explain phenomena that Isaac Newton's canon could not. Satisfying such demands was difficult, because the difference between the two models could only be apparent under extreme conditions. There seemed little hope that any terrestrial experiment could decide between them, but Einstein later identified three astronomical tests. The first was the proper calculation of the orbit of the planet Mercury—a feat that was beyond Newtonian physics (see "In Pursuit of Vulcan" in the September-October 1994 American Scientist). The second test required the comparison of light emitted from atoms in the Sun with light from similar atoms on Earth—relativity predicted that the Sun's light would have a longer wavelength (an example of the so-called redshift). The third test posited that if relativity was true, then rays of starlight that passed near the Sun would be bent compared to the same rays when the Sun was elsewhere in the sky. In each case, the relativistic effects are caused by gravity from the Sun's huge mass.
Early attempts to perform these tests did not silence Einstein's critics, because some observations supported his theory and others did not. Thus, the general theory of relativity yielded a much better solution to the Mercury problem than did Newtonian models, but another prediction of relativity, the redshift of the solar spectrum, could not be verified. (Eventually, astrophysicists learned that several other factors complicated the observation of this phenomenon.) So with one result in favor and another in doubt, the third test became something of a deciding vote for or against relativity.
Einstein first suggested how this light-bending effect could be measured in 1911. He predicted that those rays of starlight that passed closest to the Sun would be deflected by 0.85 arcseconds (0.00023 degree) because of the Sun's gravitational field. However, stars that appear next to the Sun are only visible during a total solar eclipse. To test Einstein's hypothesis, one would have to take photographs during an eclipse that showed background stars near the Sun's disk and compare them with photos taken months earlier or later, when the same stars rose in the night sky. Did stars appearing on opposite sides of the Sun's disk maintain the same spacing when the Sun was gone, or not?
This prediction seemed easy to check. Many pictures of solar eclipses already existed, as did photos of the night sky. Even so, skepticism about Einstein's theory was so prevalent that few astronomers rushed to their archives. And when they did examine previous photographs of solar eclipses, they found that the pictures were unsuited to proving or disproving Einstein's claim: The telescopes had been set to track the Sun's motion across the sky, not the stellar motions, and the slight differences between these perspectives obscured the small, predicted shifts in star positions. However, as time went by and other experiments gave equivocal results, the solar-eclipse experiment represented the best chance to test the truth of relativity.
Hoping for a Dark Noon
As early as 1912 it seemed possible to capture the necessary photographs with little fuss. In October of that year, a total solar eclipse was to run across the northern parts of South America, and the astronomical observatory of Córdoba in central Argentina was near enough to mount an expedition. Unhappily, almost all of South America was under clouds that day.
Another suitable eclipse loomed in August 1914, running northwest to southeast across eastern Europe. Erwin Freundlich, a young German astronomer, was determined to test Einstein's theory but encountered grave difficulty raising money for the trip. The scientific establishment in Germany was uninterested in paying for it, leading Einstein himself to offer his own none-too-abundant finances. With so few options, Freundlich appealed to other countries for collaborators that would help fund the expedition. He had only one taker: William Wallace Campbell and a team from the Lick Observatory in California. Later, the Berlin Academy provided additional support.
The eclipse was due August 21, but the team of Germans and Americans established a camp near Kiev well before that date to prepare for the event. Unfortunately, history intervened: On August 1, 1914, Germany declared war on Russia, and the German astronomers were taken prisoner. Russian forces expelled the older scientists and held the younger ones as prisoners of war. The Russians did allow the Americans to stay for the eclipse, but again the sky was totally clouded out. Campbell later wrote "I never knew before how keenly an eclipse astronomer feels his disappointment through clouds. One wishes that he could come home by the back door and see nobody."
The next year, at the height of the First World War, Einstein published his general theory of relativity. This timing greatly complicated the theory's dissemination because German scientific journals were then unavailable to the English-speaking world. It was an astronomer from neutral Holland who brought word of the new theory to Britain. Moreover, Britain was going through a period of almost hysterical opposition to all things German. Ardently opposed to this mindless, pervasive hatred, a young British astrophysicist named Arthur Stanley Eddington stood almost alone. Eddington was not only a rising star in astronomy but a Quaker—a religious pacifist. As such, he refused to fight in the war, although he was willing to risk his life providing aid to civilians caught in the violence. Because of his beliefs, Eddington lived on the verge of imprisonment during much of the war and suffered vicious attacks for his pacifism and efforts to counter his peers' nationalistic hostility toward German science.
Eddington learned of Einstein's general theory from the Dutch astronomer Willem de Sitter and was immediately taken with it. He was almost certainly the first (and, for a while, the only) English-speaker to understand the theory and appreciate its significance. Eddington grasped the fact that Einstein's new work meant that the eclipse experiment was an even more significant test of relativity—the general theory predicted twice as much deflection of light rays passing the Sun as did the special theory. Another suitable eclipse would occur in 1919, and although in 1915 there was no immediate hope for peace, the British Astronomer Royal, Frank Dyson, began to lay plans (no doubt at Eddington's prompting) for an expedition to photograph the event. Eddington, of course, was eager to lead such an expedition but worried that his uncertain standing with the authorities might cause difficulties for the project. Then, in a stroke of genius, Dyson wrote a carefully worded letter to officialdom. In response, the government notified Eddington that he was lucky so far in having avoided prison, and that his only hope of remaining that way was to lead Dyson's expedition, whether Eddington liked it or not! Eddington dutifully bowed to the hoped-for ultimatum.
Partly Cloudy
Around the same time, an eclipse in the United States in June 1918 was almost entirely obscured by clouds, but Campbell's team did get some photographs. These poorly exposed plates seemed to indicate no relativistic effects, much to the delight of Einstein's skeptics, including Campbell.
The eclipse of May 29, 1919, was to start near the border between Chile and Peru, then traverse South America, cross the Atlantic Ocean and arc down through central Africa. No part of the path was far from the equator, and the desirable, longest-lasting portion was in the Atlantic, a few hundred miles from the coast of Liberia. The British planners decided that the tiny island of Principe, nestled in the crook of Africa's Gulf of Guinea, would be best despite the poor astronomical viewing from low-lying tropical regions. The choice of Principe introduced other challenges. One modern travel agency advises prospective visitors to the island that "It's best to go between June and September. The rest of the year is muggy and hot—you'll be swimming in rain and your own sweat." Just in case Principe was cloudy at the crucial time, the British sent a second expedition to observe the eclipse from Sobral, in eastern Brazil.
The main instruments at both sites were existing astrographic telescopes of 33-centimeter aperture designed specifically for photographing star positions with high precision. Although these telescopes were designed to automatically follow the stars, their temporary emplacement in the field required each telescope to be immobilized as a clockwork-driven flat mirror tracked across the sky and fed light to the main lens. As an afterthought, the Brazil contingent added a small 10-centimeter telescope to its roster. In the end, it saved the day.
The expeditionaries set out months ahead of the eclipse to allow for travel difficulties. Although the war officially ended in November 1918, chaos continued for months thereafter. Upon arrival, they had to evaluate the terrain, choose a site, and set up and test their equipment. Eddington's group arrived at Principe in late April and, amid the heat and rain, found themselves under such constant attack by biting insects that they needed to work under mosquito netting most of the time. The rain grew worse as May advanced, and the day of the eclipse began with a tremendous storm. The rain stopped as the day wore on, but the totality phase of the eclipse would start at 2:15 p.m. and last only five minutes. Eddington wrote:
About 1.30 when the partial phase was well advanced, we began to get glimpses of the Sun, at 1.55 we could see the crescent (through the cloud) almost continuously and large patches of clear sky appearing. We had to carry out our programme of photographs in faith. I did not see the eclipse, being too busy changing plates, except for one glance to make sure it had begun.... We took 16 photographs ... but the cloud has interfered very much with the star images.
The weather in Brazil was much better—beautifully clear, in fact. The observers took 19 photos with the astrograph and eight with the small telescope. But when the photographs were developed, they found that despite their precautions, the astrograph's pictures showed, according to Dyson, "a serious change of focus, so that, while the stars were shown, the definition was spoiled." Even under ideal conditions, the predicted relativistic displacement on the photographs was only 1/60 of a millimeter—about a quarter of the diameter of a star on a sharply exposed image. Although they could measure such a minute shift, the poor focus made this task nearly impossible. By contrast, the small telescope's photographs were clear and sharp, but on a reduced scale.
Weighing the Data
Many months later, back in England, Eddington pondered the inconsistent results. Einstein's theory predicted a displacement of 1.75 arcseconds, but none of the experiments was in perfect agreement with the theory. The usable photos from Principe showed an average difference of 1.61±0.30 arcseconds, the astrograph in Brazil indicated a deflection of about 0.93 arcseconds (depending on how one weighted the individual spoiled photos), and the little 10-centimeter telescope gave a result of 1.98±0.12 arcseconds. The smaller device, in addition to yielding the most precise data, afforded a wider field of view and supported Einstein's theory of how the displacement should vary with angular distance from the edge of the Sun. But the validation of relativity required exact measurements, particularly because physicists had realized that Newtonian theory alone could predict a stellar displacement that was half that of Einstein's, or about 0.83 arcseconds.
Eventually, Eddington, after much discussion with Dyson, suggested an overall measurement of 1.64 arcseconds, which he took to be in pretty good agreement with Einstein, but he also gave the separate results from each telescope so others might weight them as they saw fit. Moreover, Dyson offered to send exact contact copies of the original photographic glass plates to anyone who wished to make their own measurements, which should have gone far to refute the occasional allegation that Eddington had cooked the results.
Ironically, confirmation of Eddington's conclusion (and the theory of relativity) came from Campbell's team at an eclipse in Australia in 1922, for which they determined a stellar displacement of 1.72±0.11 arcseconds. Campbell had been open in his belief that Einstein was wrong, but when his experiment proved exactly the opposite, good scientist that he was, Campbell immediately admitted his error and never opposed relativity again.
Acknowledgment
I am indebted to Dr. Jeffrey Crelinsten for granting access to his unpublished work on this topic and for providing comments on an earlier version of this article.
Bibliography
J. Donald Fernie
This year we celebrate the centenary of Albert Einstein's special theory of relativity. Indeed, 1905 was the year in which Einstein first gave notice of his astonishing abilities. He was but 26 and had just earned his doctorate, but that year he published four papers on separate topics, each of which marked a major advance in physics. The first of these, on the photo-electric effect (the subject of Roald Hoffmann's Marginalium in the previous issue), would bring him the Nobel Prize, but it was the third, on special relativity, that made him both famous and controversial. A decade after this flurry of papers, in 1915, he unveiled the theory of general relativity, shaking again the foundations of science.
To test predictions of Einstein's theories... So different was relativity from the prevailing beliefs that most physicists demanded proof that it could explain phenomena that Isaac Newton's canon could not. Satisfying such demands was difficult, because the difference between the two models could only be apparent under extreme conditions. There seemed little hope that any terrestrial experiment could decide between them, but Einstein later identified three astronomical tests. The first was the proper calculation of the orbit of the planet Mercury—a feat that was beyond Newtonian physics (see "In Pursuit of Vulcan" in the September-October 1994 American Scientist). The second test required the comparison of light emitted from atoms in the Sun with light from similar atoms on Earth—relativity predicted that the Sun's light would have a longer wavelength (an example of the so-called redshift). The third test posited that if relativity was true, then rays of starlight that passed near the Sun would be bent compared to the same rays when the Sun was elsewhere in the sky. In each case, the relativistic effects are caused by gravity from the Sun's huge mass.
Early attempts to perform these tests did not silence Einstein's critics, because some observations supported his theory and others did not. Thus, the general theory of relativity yielded a much better solution to the Mercury problem than did Newtonian models, but another prediction of relativity, the redshift of the solar spectrum, could not be verified. (Eventually, astrophysicists learned that several other factors complicated the observation of this phenomenon.) So with one result in favor and another in doubt, the third test became something of a deciding vote for or against relativity.
Einstein first suggested how this light-bending effect could be measured in 1911. He predicted that those rays of starlight that passed closest to the Sun would be deflected by 0.85 arcseconds (0.00023 degree) because of the Sun's gravitational field. However, stars that appear next to the Sun are only visible during a total solar eclipse. To test Einstein's hypothesis, one would have to take photographs during an eclipse that showed background stars near the Sun's disk and compare them with photos taken months earlier or later, when the same stars rose in the night sky. Did stars appearing on opposite sides of the Sun's disk maintain the same spacing when the Sun was gone, or not?
This prediction seemed easy to check. Many pictures of solar eclipses already existed, as did photos of the night sky. Even so, skepticism about Einstein's theory was so prevalent that few astronomers rushed to their archives. And when they did examine previous photographs of solar eclipses, they found that the pictures were unsuited to proving or disproving Einstein's claim: The telescopes had been set to track the Sun's motion across the sky, not the stellar motions, and the slight differences between these perspectives obscured the small, predicted shifts in star positions. However, as time went by and other experiments gave equivocal results, the solar-eclipse experiment represented the best chance to test the truth of relativity.
Hoping for a Dark Noon
As early as 1912 it seemed possible to capture the necessary photographs with little fuss. In October of that year, a total solar eclipse was to run across the northern parts of South America, and the astronomical observatory of Córdoba in central Argentina was near enough to mount an expedition. Unhappily, almost all of South America was under clouds that day.
Another suitable eclipse loomed in August 1914, running northwest to southeast across eastern Europe. Erwin Freundlich, a young German astronomer, was determined to test Einstein's theory but encountered grave difficulty raising money for the trip. The scientific establishment in Germany was uninterested in paying for it, leading Einstein himself to offer his own none-too-abundant finances. With so few options, Freundlich appealed to other countries for collaborators that would help fund the expedition. He had only one taker: William Wallace Campbell and a team from the Lick Observatory in California. Later, the Berlin Academy provided additional support.
The eclipse was due August 21, but the team of Germans and Americans established a camp near Kiev well before that date to prepare for the event. Unfortunately, history intervened: On August 1, 1914, Germany declared war on Russia, and the German astronomers were taken prisoner. Russian forces expelled the older scientists and held the younger ones as prisoners of war. The Russians did allow the Americans to stay for the eclipse, but again the sky was totally clouded out. Campbell later wrote "I never knew before how keenly an eclipse astronomer feels his disappointment through clouds. One wishes that he could come home by the back door and see nobody."
The next year, at the height of the First World War, Einstein published his general theory of relativity. This timing greatly complicated the theory's dissemination because German scientific journals were then unavailable to the English-speaking world. It was an astronomer from neutral Holland who brought word of the new theory to Britain. Moreover, Britain was going through a period of almost hysterical opposition to all things German. Ardently opposed to this mindless, pervasive hatred, a young British astrophysicist named Arthur Stanley Eddington stood almost alone. Eddington was not only a rising star in astronomy but a Quaker—a religious pacifist. As such, he refused to fight in the war, although he was willing to risk his life providing aid to civilians caught in the violence. Because of his beliefs, Eddington lived on the verge of imprisonment during much of the war and suffered vicious attacks for his pacifism and efforts to counter his peers' nationalistic hostility toward German science.
Eddington learned of Einstein's general theory from the Dutch astronomer Willem de Sitter and was immediately taken with it. He was almost certainly the first (and, for a while, the only) English-speaker to understand the theory and appreciate its significance. Eddington grasped the fact that Einstein's new work meant that the eclipse experiment was an even more significant test of relativity—the general theory predicted twice as much deflection of light rays passing the Sun as did the special theory. Another suitable eclipse would occur in 1919, and although in 1915 there was no immediate hope for peace, the British Astronomer Royal, Frank Dyson, began to lay plans (no doubt at Eddington's prompting) for an expedition to photograph the event. Eddington, of course, was eager to lead such an expedition but worried that his uncertain standing with the authorities might cause difficulties for the project. Then, in a stroke of genius, Dyson wrote a carefully worded letter to officialdom. In response, the government notified Eddington that he was lucky so far in having avoided prison, and that his only hope of remaining that way was to lead Dyson's expedition, whether Eddington liked it or not! Eddington dutifully bowed to the hoped-for ultimatum.
Partly Cloudy
Around the same time, an eclipse in the United States in June 1918 was almost entirely obscured by clouds, but Campbell's team did get some photographs. These poorly exposed plates seemed to indicate no relativistic effects, much to the delight of Einstein's skeptics, including Campbell.
The eclipse of May 29, 1919, was to start near the border between Chile and Peru, then traverse South America, cross the Atlantic Ocean and arc down through central Africa. No part of the path was far from the equator, and the desirable, longest-lasting portion was in the Atlantic, a few hundred miles from the coast of Liberia. The British planners decided that the tiny island of Principe, nestled in the crook of Africa's Gulf of Guinea, would be best despite the poor astronomical viewing from low-lying tropical regions. The choice of Principe introduced other challenges. One modern travel agency advises prospective visitors to the island that "It's best to go between June and September. The rest of the year is muggy and hot—you'll be swimming in rain and your own sweat." Just in case Principe was cloudy at the crucial time, the British sent a second expedition to observe the eclipse from Sobral, in eastern Brazil.
The main instruments at both sites were existing astrographic telescopes of 33-centimeter aperture designed specifically for photographing star positions with high precision. Although these telescopes were designed to automatically follow the stars, their temporary emplacement in the field required each telescope to be immobilized as a clockwork-driven flat mirror tracked across the sky and fed light to the main lens. As an afterthought, the Brazil contingent added a small 10-centimeter telescope to its roster. In the end, it saved the day.
The expeditionaries set out months ahead of the eclipse to allow for travel difficulties. Although the war officially ended in November 1918, chaos continued for months thereafter. Upon arrival, they had to evaluate the terrain, choose a site, and set up and test their equipment. Eddington's group arrived at Principe in late April and, amid the heat and rain, found themselves under such constant attack by biting insects that they needed to work under mosquito netting most of the time. The rain grew worse as May advanced, and the day of the eclipse began with a tremendous storm. The rain stopped as the day wore on, but the totality phase of the eclipse would start at 2:15 p.m. and last only five minutes. Eddington wrote:
About 1.30 when the partial phase was well advanced, we began to get glimpses of the Sun, at 1.55 we could see the crescent (through the cloud) almost continuously and large patches of clear sky appearing. We had to carry out our programme of photographs in faith. I did not see the eclipse, being too busy changing plates, except for one glance to make sure it had begun.... We took 16 photographs ... but the cloud has interfered very much with the star images.
The weather in Brazil was much better—beautifully clear, in fact. The observers took 19 photos with the astrograph and eight with the small telescope. But when the photographs were developed, they found that despite their precautions, the astrograph's pictures showed, according to Dyson, "a serious change of focus, so that, while the stars were shown, the definition was spoiled." Even under ideal conditions, the predicted relativistic displacement on the photographs was only 1/60 of a millimeter—about a quarter of the diameter of a star on a sharply exposed image. Although they could measure such a minute shift, the poor focus made this task nearly impossible. By contrast, the small telescope's photographs were clear and sharp, but on a reduced scale.
Weighing the Data
Many months later, back in England, Eddington pondered the inconsistent results. Einstein's theory predicted a displacement of 1.75 arcseconds, but none of the experiments was in perfect agreement with the theory. The usable photos from Principe showed an average difference of 1.61±0.30 arcseconds, the astrograph in Brazil indicated a deflection of about 0.93 arcseconds (depending on how one weighted the individual spoiled photos), and the little 10-centimeter telescope gave a result of 1.98±0.12 arcseconds. The smaller device, in addition to yielding the most precise data, afforded a wider field of view and supported Einstein's theory of how the displacement should vary with angular distance from the edge of the Sun. But the validation of relativity required exact measurements, particularly because physicists had realized that Newtonian theory alone could predict a stellar displacement that was half that of Einstein's, or about 0.83 arcseconds.
Eventually, Eddington, after much discussion with Dyson, suggested an overall measurement of 1.64 arcseconds, which he took to be in pretty good agreement with Einstein, but he also gave the separate results from each telescope so others might weight them as they saw fit. Moreover, Dyson offered to send exact contact copies of the original photographic glass plates to anyone who wished to make their own measurements, which should have gone far to refute the occasional allegation that Eddington had cooked the results.
Ironically, confirmation of Eddington's conclusion (and the theory of relativity) came from Campbell's team at an eclipse in Australia in 1922, for which they determined a stellar displacement of 1.72±0.11 arcseconds. Campbell had been open in his belief that Einstein was wrong, but when his experiment proved exactly the opposite, good scientist that he was, Campbell immediately admitted his error and never opposed relativity again.
Acknowledgment
I am indebted to Dr. Jeffrey Crelinsten for granting access to his unpublished work on this topic and for providing comments on an earlier version of this article.
Bibliography
- Clark, R. W. 1971. Einstein: The Life and Times. New York: World Publishing Company.
- Crelinsten, J. In press. Einstein's Jury: The Race to Test Relativity. Princeton, New Jersey: Princeton University Press.
- Stanley, M. 2003. An expedition to heal the wounds of war: The 1919 eclipse and Eddington as Quaker adventurer. Isis 94:57-89.
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