A novel to end all novels
James Joyce once claimed that he has put in his book “Ulysses” so many enigmas and puzzles that “it will keep the professors busy for centuries arguing over what I meant, and that’s the only way of insuring one’s immortality”. It is doubtful whether Joyce had in mind professors of Mathematics but it seems that indeed many tiny enigmas and puzzles relevant to Mathematics and Science are present in “Ulysses”. Leopold Bloom is the central character of the so called “novel to end all novels”, a more than 700 page long book. The whole plot, loosely connected to Homer’s “Odyssey”, unfolds in a single day, June 16, 1904, and follows Bloom’s day long odyssey in the city of Dublin, his wanderings, his encounters, his memories and his thoughts in the form of an “internal monologue”. This technique reveals the character’s thoughts, expressed in first person and unprocessed, often without apparent cohesion. Bloom is a rather complex character, haunted by the death of his infant son and the affair of his wife Molly, an opera singer, with her manager. The text itself is of immense complexity, frequent ambiguities and presents a patchwork of several different styles of expression in different parts.
A little finger blotting out the sun’s disk
In the “Lestrygonians” chapter and during one of his wanderings in Dublin, Bloom stands before the window of “Yeates and Son”, an optical store. Triggered by what he sees, his mind strays from cooking and food to his old glasses that need to get fixed, the Germans who are “making their way everywhere”, including Goerz lenses, and the possibility of getting a new pair of glasses at the railway lost property office.
“Astonishing the things people leave behind in trains and cloak rooms. What do they be thinking about? Women too. Incredible. Last year travelling to Ennis had to pick up that farmers’ daughter’s bag and hand it to her at Limerick station. Unclaimed money too.”
And then Bloom suddenly starts thinking of experiments and even carries out a couple:
“There’s a little watch up there on the roof of the bank to test those glasses by.
His lids came down on the lower rims of his irides. Can’t see it. If you imagine it’s there you can almost see it. Can’t see it.
He faced about and, standing between the awnings, held out his right hand at arm’s length toward the sun. Wanted to try that often. Yes: completely. The tip of his little finger blotted out the sun’s disk.”
What Bloom is doing is exactly what painters do when using their thumb or a pencil to measure the relative size of their subject. The subject here is the sun and Bloom confirms his suspicion that its apparent size is so small that a little finger at arm’s length could completely cover the solar disk. Anyone who has tried to take a photograph of a person with a magnificent sunset in the background must have arrived, often with surprise, to similar conclusions: eventually the sun appears tiny in the picture and the sunset not so magnificent. So what is the apparent size of the sun after all and how is it measured?
Measuring the solar disk
Any object of any size, no matter how small, can completely cover the solar disk when placed at a small enough distance from our eyes (this is exactly what the brim of a hat does). However, for every specific object there is a specific distance for which the object just barely covers the sun. For example, a soccer ball (which has a diameter of approximately 22 cm) just barely covers the sun when put at a distance of 25 m from the observer. In this case we can say that the ball and the sun have the same apparent size i.e. they look the same although their actual sizes are very different. An orange has the same apparent size as the sun when put approximately 10 m from the observer. These examples express well the fact that, perhaps against our intuition, the apparent size of the sun is actually very small. One can convince himself or herself by trying to hit with a stone an orange located at a distance of 10 m : the target seems hopelessly small. These conclusions can be generalized for any object of size s cm. It can be seen that such an object has the same apparent size as the sun when placed at a characteristic distance of about 110s cm from the observer. Therefore a pinky finger should indeed have the same apparent size as the sun, and thus blot out the sun’s disk, when placed at a distance from the eye of the observer 110 times its width: a requirement obviously fulfilled by any individual of reasonable body proportions. For example, a finger 2 cm in width should be placed at a distance greater than 2,2m when the hand is held out, in order not to completely cover the solar disk. Obviously, the bigger an object is, the further away it must be placed in order to have the same apparent size as the sun. Then, it takes approximately 720 such objects to complete a circle with radius the characteristic distance 110s m. For example it takes 720 soccer balls to complete a full circle with a radius of 25 m. It also takes 720 oranges to complete a full circle with a radius of 10 m.
Aristarchus and Thales
The ancient Greek scientists had arrived to similar conclusions. In his ingenious work “Psammites”, the ancient Greek scientist and inventor Archimedes (ca 287 – 212 B.C.) refers to the astronomer and mathematician Aristarchus (310 -230 B.C.) who has “...found that the sun appears as the seven hundred and twentieth part of the circle of the zodiac“. And in his work “Βίοι και γνώμαι των εν φιλοσοφία ευδοκιμησάντων” (Life and opinions of the distinguished in philosophy), Diogenes Laertius (probably third century A.C.), a biographer of ancient Greek scientists, states that the famous Thales of Miletus (ca 624 – 546 BC) is “mentioned by some as the first to determine the size of the sun to be the one seven hundred and twentieth of the solar circle“. In other words it takes 720 suns to complete a full circle in the horizon and therefore, we conclude, it takes 720 objects with apparent size equal to the sun’s to complete a full circle with radius the distance of these objects from the observer. Taking in account the well known elementary school fact that a full circle consists of 360 degrees, it is obvious that the apparent size of the sun in the sky is approximately 360/720=0,5 degrees. Due to the elliptic orbit of the earth around the sun this angle actually varies slightly having an average value of 0.533 degrees. This result is quite surprising: half a degree is the measure of the angle made by the minute hand of an analogue watch in just 5 seconds. It is an angle so small that the respective motion of the minute hand is unperceivable to the human eye. To understand this better one may consider herself or himself standing at the centre of a huge, imaginary clock with its face on the ground and its hands extending to the horizon, as far as human eyes can see. Each second, the second hand of such a clock moves defining a certain angle. Within this angle on the horizon line, there is enough space to accommodate 12 solar disks, next to each other. It is extraordinary that the apparent size of a star the presence of which we perceive so bright and imposing in the sky, is after all expressed in almost insignificant subdivisions of a unit so small as a degree.
The coincidental eclipse
By a remarkable coincidence (and also quite brief moment in astronomical time scale as distances slowly change), the average distance of the moon from the earth is aproximately 110 times the lunar diameter, just as the average distance of the sun from the earth is 110 times the solar diameter. Therefore, the apparent size of the moon is approximately equal to the apparent size of the sun. In fact the difference between the average apparent sizes of the sun and moon is less than 2%. Sometimes, under certain circumstances, the moon happens to align between the sun and the earth and then it casts on the earth’s surface a circular shadow, travelling with an ultrasonic speed, and drawing thus a line several thousand kilometers long but just a few kilometers wide. From within this shadow we can observe an eclipse of the sun, during which the lunar disk fully covers the sun and creates night conditions for a few minutes. If at the moment of alignment the moon happens to be approximately at its minimum distance from the earth, then the eclipse is total and the solar disk is fully covered. This impressive phenomenon, which owes its occurence to a pure coincidence, is rather rare: it is observed on average two or three times a year from various locations on earth. According to Diogenes Laertius, Thales of Miletus had also realized that the apparent sizes of the sun and moon are approximately equal, probably due to the observation that the lunar disk can fully cover the solar disk during a total eclipse.
Returning to “Ulysses”, after observing that a little finger can indeed blot out the solar disk, Leopold Bloom proceeds by connecting his experiment to the solar eclipses:
“If I had black glasses. Interesting. There was a lot of talk about those sunspots when we were in Lombard street west. Terrific explosions they are. There will be a total eclipse this year: autumn some time“.
It is rather uncertain to what eclipse Bloom refers too, as there were no total solar eclipses visible from the British Islands between 1724 and 1925. There was however a total eclipse in September 9, 1905 across the Pacific Ocean, its peak right in the middle of the ocean. Remarkably, right in front of the optical store, Bloom moves his thoughts from the easy to understand apparent sizes and eclipses to a question that makes him quit immediatelly. “What’s parallax?” he wonders and then resigns: “Ah. His hand fell again to his side. Never know anything about it. Waste of time. Gasballs spinning about, crossing each other, passing. Same old dingdong always. Gas, then solid, then world, then dead shell shifting around, frozen rock like that pineapple rock.” As a matter of fact, Bloom observed parallax just a minute ago, the moment he closed one eye in order to observe his little finger blotting out the sun’s disk. Closing either eye the finger appears to move to a different position, as it is observed from slightly different viewpoints (the positions of the two eyes). The position of the finger and those of the eyes define an angle with the vertex at the finger’s position. Parallax in this case is half of this angle. Actually parallax is applicable in quite a number of similar situations, ranging from computer graphics to measuring the distances of stars.
James Joyce has used several different ways of describing what exactly “Ulysses” is after all, before settling to rather bluntly naming it “a book“. “Epic“, “maledettisimo romanzaccione” (dog Italian for “hell of a big novel”) are other terms used by him, playfully or not. From a certain perspective, another of his choices, “encyclopaedia”, often seems quite fitting for “Ulysses”. In a parallactic manner, “Ulysses” appears different from different viewpoints: often an encyclopaedia of history, mythology, literature or religion and even, sometimes, an encyclopaedia of science.