James Joyce himself once claimed that he has put in “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 rather uncertain whether Joyce had in mind professors of Mathematics but it seems that "Ulysses" is actually packed with tiny enigmas and puzzles relevant to Mathematics and Science. 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 wandering, 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 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 and frequent ambiguities while adopting at the same time several different styles of expression in different parts. Often the reader is presented with such difficulties that comprehension seems beyond grasp.
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 Goetz 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 does here is exactly what painters do when using their thumb or a pencil to measure the relative size of their subject. And his subject is the sun. Bloom confirms his suspicion that the apparent size of the sun 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: in the end 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?
Any object of any size, no matter how small, can completely cover the solar disk, if it is located at a distance small enough from our eyes (this is exactly what the brim of a hat does). However, for every specific object there is a specific distance such as 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 put at a characteristic distance of about 110s m from the observer. Obviously, the bigger an object is, the further away it must be put 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.
The ancient Greek scientists had arrived to similar conclusions. 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 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. This result is amazing: half a degree is the measure of the angle made by the minute hand of an analogue watch in just 5 seconds. This angle is so small that the respective motion of the minute hand is unperceivable to the human eye. To understand this better one may imagine himself or herself 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 our sun, a star the presence of which we perceive so bright and imposing in the sky, is after all measured using almost insignificant subdivisions of a unit so small as a degree.
However, it must be noted that the apparent size of the sun is not constant. Due to the elliptic shape of the earth's orbit around the sun, sometimes the earth comes a little closer and sometimes goes a little further away from the sun. The result is a slight variation, about 3%, of the sun’s apparent size during the year. Its average value has been determined to be 0,533 degrees.
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 Goetz 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 does here is exactly what painters do when using their thumb or a pencil to measure the relative size of their subject. And his subject is the sun. Bloom confirms his suspicion that the apparent size of the sun 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: in the end 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?
Any object of any size, no matter how small, can completely cover the solar disk, if it is located at a distance small enough from our eyes (this is exactly what the brim of a hat does). However, for every specific object there is a specific distance such as 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 put at a characteristic distance of about 110s m from the observer. Obviously, the bigger an object is, the further away it must be put 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.
The ancient Greek scientists had arrived to similar conclusions. 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 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. This result is amazing: half a degree is the measure of the angle made by the minute hand of an analogue watch in just 5 seconds. This angle is so small that the respective motion of the minute hand is unperceivable to the human eye. To understand this better one may imagine himself or herself 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 our sun, a star the presence of which we perceive so bright and imposing in the sky, is after all measured using almost insignificant subdivisions of a unit so small as a degree.
However, it must be noted that the apparent size of the sun is not constant. Due to the elliptic shape of the earth's orbit around the sun, sometimes the earth comes a little closer and sometimes goes a little further away from the sun. The result is a slight variation, about 3%, of the sun’s apparent size during the year. Its average value has been determined to be 0,533 degrees.
