ࡱ> +-()*0125@ ݬbjbj22 ,XX%84 Lr5\ L " !!!4444444$(6Rz8v5[+!![+[+5 ,5e4e4e4[+  4e4[+4e4 e444 P 0ZVNCqk1 44B50r54838484l!Y$e4G&'!!!554d 4X4THE BRIDGES OF KINGS HILL Vithal Rajan vithalrajan(at)hotmail.com A violet autumnal evening had fallen over the city of Isfahan. Sofiyas house was in the northern outskirts, and her room was the northernmost, the casements flung open to let in the scented breeze from her garden. She snuggled deep under the thick quilt, with riotous summer flowers hand-embroidered by her mother, and looked at the far-off mountains, sprinkled with fresh snow. Daddy! Wouldnt it be lovely to live high up on the mountains? she exclaimed. In a city built on the highest peak? Naseeruddin wasnt so sure. It could get frightfully cold, darling, he said. And think of sleet on all the steep streets! Everyone would be sliding and slipping, and poor Grandma would get hurt. Even a donkey cart would find it difficult coming up with the morning groceries. Sofiya was still enthusiastic. But Daddy, it would all be so much fun! And Im sure I could hold Grandmas arm when we went walking up and down the streets, she said. Why dont people ever live on mountains? But they do, my dear, said Naseeruddin reasonably. Theres Lhasa, thats the capital of Tibet, and theres Quito, thats the capital of Ecuador. But these cities are in high mountain valleys, sheltered by the mountains themselves. In the old days, kings built their palaces within forts on hills to protect them from raiders. In fact in those mountains you see to the north, but far, far to the east, near where they join the Hindu Kush, a king once built a new city, which everyone called the Kings Hill. He got into lots of trouble till he was helped by a very poor man. Sofiya was interested. Oh, do tell me about him, she said, settling back comfortably on her pillows. All right, said Naseeruddin, Ill tell you a story my uncle the Mullah told me long ago, but you must go to sleep after the story. Sofiya nodded, and turned to look at the mountains once again, now touched with russet and gold by the fading sun. Long, long ago, in the year Minus 1234, which is very long ago, far to the east in Persia started Naseeruddin in his best story-telling style there lived a king called well, we dont know his real name but he built his capital on a hill, and it was called Kings Hill. He wanted to protect himself from nomads and other raiders, especially the Bruzz, now dont ask me why they were called the Bruzz. Maybe because anyone who messed with them got bruised. Thats probably the right explanation, for the Head of the Bruzz was called The Bruiser! Anyway, maybe they later went on into Lebanon, and came to be known as the Druss! Such name changes happened all the time in olden times. The Hindus, for example, were people who lived along the Sindhu river, but the Greeks got their name wrong, and they are stuck with it now, poor fellows. Their Banjara tribes came into Persia along with Chengiz Khans troops, and then followed them into Egypt. But years later when these nomads went to Roumania, they were called Gypsies, because everyone thought they really belonged to Egypt, and later when they lifted their tents and went to the broad pasture lands of Hungary, the Europeans called them Romany! So, you see, names change with the years, or they used to. Anyway, the king wanted to make peace with the Bruzz, so he married the Bruzi princess, and called her His Woman, they were quite crude in those days, you know, in fact the French still do so and the Bruzz not to be out-done called him Her Man! So thats the only name we know him by Herman since an account of those days has been left to us by the Bruzi princess who was a lot cleverer than the king. Now the king had his good points, though he couldnt read or write, for one thing, he danced beautifully. Thats what really won over the Bruzi princess, if you want to know. He invented most of the dances we know, except the waltz, which was taught to him by his nurse, Matilda. His favourite dance, though, was the Salsa. You thought it was invented by Cubans, didnt you? Well, you are wrong, as is everybody else, it was taken to Cuba by descendents of the people of Kings Hill. It was first danced by Herman of the Salsa, as he came to be known later, up and down the land. Herman of the Salsa had many knights in his entourage, in fact, more than many kings of his day. But they all had duties to perform guarding the city from raiding Bruzz, or supervising harvesting, or even just chivvying the peasants so that they wouldnt doze off under the balmy summer sun. While Herman was glad his knights were doing their duty, he got impatient when it came to dancing time, and there were very few around when a good tune was struck up. So he invented a special Two-Tone Bell, which all could hear, even if they were at the edges of his kingdom. The best dancing knights were to hurry back to the palace for a dance when the Two-Tone Bell was struck under pain of royal displeasure. When these Two-Tone Knights for thats how the ordinary people came to call them came hurrying onto the dance floor, the king was very pleased, and would salsa away the night with his queen. The Two-Tone Knights, though, began to grumble. Not that they did not like to dance. They liked nothing better, that is, besides chivvying peasants or killing the Bruzz. But getting into the palace was a big problem. The king like a prudent ruler had chosen a hilly island in a river for his capital, so that the river formed a natural protective moat all round it. His people also liked the idea very much for the river was full of eels, which they could fish straight from their living room windows. But once the city was built, they started calling the eel river The Pretzel, dont ask me why, for there are no pretzels in Persia, people are just strange. The king had a wooden drawbridge thrown over the river which could be pulled back quickly if there was an attack. But the knights found this single bridge a nuisance. They had to splash through the river wherever they were coming from, and go all the way round to find it. The people also didnt like all these Two-Tone Knights clogging up the traffic as they thundered over the simple wooden bridge, especially on market days, but they were more polite about their grievance. All right, we will have another bridge on the other side, said the king, who was all accommodation. But what about us, who are on the far northern marches, argued some knights. And yes, what about us in the southern pastures? asked others. All right, said the king, we will have bridges all round, to the east, to the west, to the north and even to the south. So that was settled, and four bridges were built. But the new bridges didnt make everyone happy. The butchers were the first to complain. Galloping knights throw a dirty big mask of dust all over the meat, they said, and it was no use the kings chef yelling at us, he should speak to the knights first. The greengrocers said, Yes, we can wash all the vegetables after all the knights have ridden off, true, but it all takes time, and a down-turn in the economy doesnt help the city, or anyone else, Your Honour. Only the blacksmiths were happy with their bridge, for hard-riding knights meant custom, so they welcomed them, dust or no dust. So, the king, who was really a good chap, built a new broad bridge with big state-supported markets on either side, which pleased all the shop-keepers, and a new high bridge that no-one but the knights would want to use, and finally to please His Woman, that is the queen, a nice little bridge leading to the flowered gardens from which bee-keepers brought her everyday a fresh jar of honey. She had a sweet tooth, besides having a sweet smile and a sweet nature. Every year in spring, she would grant a chest full of gold to anyone who brought her the sweetest honey for tea, and for that year that honey would be known as The Grantchester Honey for Tea, believe you me. Sofiya interrupted: Did you say there were seven bridges in all, Daddy? Yes, there were seven in all, said Naseeruddin. I am getting muddled, persisted Sofiya. Can you show me a map where they were in the town? Naseeruddin smiled. As it so happens, Great Uncle the Maulana, did leave me a map. Look! Here it is! And he showed Sofiya this map.  Sofiya smiled when she saw the map. I understand now everything clearly, she said, do go on, Daddy. Everyone was very, very happy with these seven bridges of Kings Hill, continued Naseeruddin, except poor old Kalin the Oiler. You see, being on a hill, the city got quite cold most nights, and everyone, especially the king and his knights in their draughty dancing halls, needed heating oil for their furnaces. Persia has always been rich in oil, so thats why all these bandits from olden days have raided the country under one pretext or another. But old Kalin the Oiler didnt have to worry about that, his job was to supply oil to Kings Hill, and he took his donkey and oil-cart laden with oil jars everyday into the city through one bridge or the other, he had a day-pass for every one of them. Once he entered through one bridge, lets say the Shopkeepers Bridge, the guards who had strict instructions would tear off his bridge-pass and tell him to use another bridge on his way out. They were all highly trained, and kept a keen eye out for anyone who did not have a pass. Even in those days, as I said, there were all these foreign terrorists who tried to steal the countrys oil. Kalin the Oiler didnt like the steep streets of the city, built as it was on a hill. His donkey liked them even less. The least tiring way to supply oil to houses in different parts of the city, and even to various wings of the kings sprawling palace, was to go in and out of the bridges, off-load his jars and be back home before nightfall. But every time he went through a bridge, his day-pass got torn off and when he had finished his last delivery he found himself back in the middle of town without any bridge-pass left to get him home. You can imagine poor Kalins state of mind. Several nights he spent at a friends home or at a lodging. The next day he would be late with the heating oil, and get roundly scolded. The Superintendent of the Bridge Guards called him a stupid old Oiler, and said surely he could manage his routes better and find his way home at night. One night, the lodging-house keeper advised him to consult the Great Philosopher, who everyone knew, knew all the answers. How do I find the Great Philosopher? asked poor old Kalin the Oiler. Well thats easy, said the lodging-house keeper. Stand in the main square at 7.43 sharp in the morning and you will meet him taking his constitutional. Can set your watch, God bless him! Kalin was in the main square by 7.15 in the morning, he was that anxious to get the Great Philosophers advice. It was a sharp cold morning, and he had to jump up and down and stamp his feet to keep warm and blow on his fingers, but as the lodging-house keeper had said, the Great Philosopher entered the square from the palace end at 7.43 sharp. He wore a long velvet coat that sparkled with stars and crescents, and on his head was a high black fur cap. He waved his arms and talked to himself in deep tones, no doubt about very weighty matters. Excuse me, SirI mean, Oh, Great Philosopher, stammered poor old Kalin the Oiler, I dont mean to disturb your Highness, no Sir, but but could you show me the way to cross the seven bridges just once every day, and still get home? The Great Philosopher looked all round at hearing his voice, and then at last spotted Kalin the Oiler standing right in front of his nose. What did you say? asked the Great Philosopher irritably. Speak clearly, man, dont mumble! Kalin the Oiler stammered out his humble request once again. Ah, Ha! cried the Great Philosopher, mightily pleased. I see you are not yet Enlightened! You must emerge from your own self-imposed immaturity! Actually, he said unmndigkeit for thats the way philosophers speak, but we shouldnt for we are ordinary folk. What you need to do my dear oily friend, said the Great Philosopher animatedly, poking Kalin in the chest with a long fore-finger, what you need to do is use your own intellect without the direction of another. Do you understand? You are responsible for this immaturity he once again said unmndigkeit and, and dependence! Its cause, permit me to inform you, is not a lack of intelligence, but a lack of determination and courage! Sapere aude! Dare to know! With that the Great Philosopher turned and started to walk away, but Kalin was quite determined to get advice, if there was advice to be got. He ran after him and grabbed his long flowing velvet coat. Please, Sir, please tell me how to cross the seven bridges, he pleaded. Shant! Wont! Cant! Cant! Cant! yelled the Great Philosopher, stamping his foot like a child. Why do you keep shouting Cant! Cant! Cant? asked Kalin in confusion. What else can I say? asked the Great Philosopher, puzzled. Jesus! exclaimed Kalin in anguish. Yes, you spoke to me? asked the Great Philosopher calmly this time. Kalin shook his head. I just said Jesus, he said weakly. I thought so, said the Great Philosopher thoughtfully. The name we commonly use to call on the Messiah. In ancient Aramaic we use another name for God with us Anyway, what did you want? Kalin looked at him helplessly. Yes, I recollect now, said the Great Philosopher smiling brightly. You, by the looks of you, should be an oiler. Right? You, in other words, belong to the category of servants known as oilers. And it is imperative you do your duty as an oiler, or we will all freeze to death of cold. There! I have solved your problem! The Great Philosopher once again turned to go. Kalin clung to him like a drowning man. Sir! Sir! Dodo tell me how I am to cross the seven bridges just once every day! he stammered out weakly. So! Thats your problem! Why didnt you say so at the start. My dear Oiler, act only according to that maxim whereby you can at the same time will that it should become a universal law! Thats all you need to know." And with another muttered unmndigkeit the Great Philosopher was gone. Never mind him, said John the herder, who had been standing by listening to all this, along with his interested swine. He isnt all that great a philosopher, as they make him out to be. Kalin went back home very dejected. But he was a game old oiler. He tried every possible route he could think off, but every time he found himself locked into the city without any way home open till next morning brought him a new set of day-passes. Everyone laughed at his stupidity. Funny stories about Kalin the Oiler circulated well beyond even the boundaries of that small kingdom. He didnt mind if people laughed at him and his stupidity. He was a good-natured chap. But he did mind spending the night in a lodging in town. So, when he really could not find a way to cross every bridge on his route without getting caught inside for the night, he carefully put away one last bridge-pass into a deep inside pocket, and finished his deliveries walking up and down the steep streets. This made him very tired, and once or twice in a day his donkey would just sit down and refuse to move. Consequently, he got home very late, tired, ill, and irritable. He smiled less and less at the jokes made up about him. Once or twice when he really got fed up at all the cracks about his slow wits, he would even mutter something back about people who thought they were very clever but would find out who was really dumber one day, if there was a God. It all came to a head when the king decided to celebrate a Day of Mirth, and called it Kalin the Oilers Day. All the streets were festooned with buntings. The pubs were opened up as Free Houses for Kalin the Oilers Day, and free beer was poured into anyone who could still stand on his feet. There was a lot of laughter and dancing the salsa on the streets. The king took the lead in the dancing and meeting Kalin the Oiler sitting morosely by a fountain in a cobbled courtyard asked him all in good humour how he and his donkey got so muddled everyday on their rounds. Kalin had had enough. If you are so clever, Your Majesty, he said, at the top of his voice, please show me yourself how the seven bridges are to be crossed, one and only one at a time! Everyone, as you can expect, was quite shocked at his disrespect to the king, and they all fell silent. But the king was in high humour, having himself visited several Free Houses that day, much to the gratification of the pub-owners, and he said laughingly, why, he would salsa his way through every bridge. And he danced down the greengrocers bridge, followed by his knights, and tradesmen, and great ladies, and parlour-maids, and old soldiers, and bank clerks, and governesses, and, and, well everyone else in town who could dance. They danced down a bridge and danced up another, till they had danced through all the bridges. Then laughing, and tired, they all sat down by the fountain in the main square while boys brought out great pints of beer for everyone. The king patted Kalin on the back and called him A Good Fellow, great ladies smiled at him, and the pub-owner ordered that three pints of beer should be set up in front of him. But Your Majesty, said Kalin stubbornly, when there was a little quiet, you crossed the Honey Bridge twice I counted so you still havent done what you said you would do! Everyone was quite shocked at his temerity. I mean it was the Day of Mirth, all right, but he was just a lowly oilier and was taking his license far too far. But the king prided himself on being a just man. Having downed some excellent draught beer he felt fresh again, and challenged everyone to join him in another round bridge dance. No one dared to call it a day, so off they went again, this time keeping careful count of the bridges they crossed. As Kalin could have told them, whichever route they took, they ended up on the wrong side. Even the king was getting tired, and a thoughtful frown sat on his royal brow. Several knights were openly cursing that meddling oiler and his cursed oil. The ladies had given up long before the shades of evening began grew over the main square. The king sat in the centre of a silent crowd, immersed in thought, gulping his beer with great gasps. You are very right, Kalin, he said at long last, There is no way the seven bridges can be crossed just once every day without getting to be on the wrong side. You have shown Us how thoughtless We have been. From this day on you shall live in a special wing of the palace. Tell you what! We will make it into an inn. We shall name it Kalins Inn! And We shall come there everyday to have a drink with you! Everyone cheered of course, partly in relief that they were done with dancing for the day. But they also liked their king and his easy ways, and they liked Kalin the Oiler as well, and met up for a drink and a dance at Kalins Inn, whenever they could get away from work, which was quite often in those days. Well, that was a long, long time ago, but the fame of Kalins Inn and the great people who gathered there came down through the ages, till these days most people just know the place as Kalin Inn town. Sofiya laughed sleepily into her pillow. Oh, Daddy! You have got everything so muddled up, its funny! she said. Naseeruddin was puzzled. Got it all muddled up? he asked with concern. What have I got muddled up? But Sofiya was already fast asleep. Naseeruddin gently shut the window. It was quite dark outside. He then kissed his daughter softly on the brow, and tiptoed out. Whatever did Sofiya mean? He shook his head mystified, and wondered if anyone could help him straighten out the story. Later, he asked around but no one could. Postscript: Every good story should have a postscript, so we know what happened later, dont you agree? When Sofiya went back to school, she left a little note for her father. This is what it said: Dearest Daddy, Great Uncle the Maulana was pulling your leg, Daddy, when he told you the story. Its really about the Seven Bridges of Knigsberg, in East Prussia, which were figured in a theorem in topology by the great 18th century mathematician, Leonhard Euler. Konigsberg was the capital of the Teutonic Knights of old, and defended against the Slav tribes of the Bruzi by Hermann von Salza. The river that flows through Knigsberg, circling the island of Kniephof, is the Pregel, into which flows the Angrapa, meaning eels in Old Prussian. Only the bridges in the story are given their right names; all the other names are close to the real ones and meant only to tease you to discover them. Oh, I think he threw in that reference to Rupert Brookes Great War poem on Grantchester, because it is across the river Cam from Cam-bridge, do you see? While there is no record that Euler sought the advice of the great philosopher Immanuel Kant of Knigsberg, it is just possible. Incidentally, Johann Herder was the only notable philosopher of those times who did not praise Kant, whose Christian name is derived from the Aramaic name for the Messiah, or God with us. Knigsberg was renamed Kaliningrad by the Russians, and is now part of Russia. I knew all this soon after you started telling the story, Daddy, but kept wondering why Great Uncle the Maulana had presented you an important mathematical puzzle of its time as a fable. I have now figured out why. You have always been scared of mathematics and grumbled that it is dry, and useful only for counting money. Mathematical discoveries are just as enchanting as any fable, Daddy, as full of lore as any epic, and on the way lead you to make amazing discoveries about life and people. The Maulana has shown you that the civic centre of this mathematical puzzle was also the home of the greatest moral philosopher of the Enlightenment, and that it had also been the city of the medieval Teutonic Knights! Euler set out to solve a popular puzzle, but on the way he wrote the first paper in topology! Anybody can understand it, Daddy, even you, so I am attaching it as an appendix. Much love to my silly old Daddy, Sofiya Naseeruddin turned to the appendix and for his daughters sake studied it for all of ten minutes. Then, when he was about to see that he could understand what it was all about, he put it aside with a yawn. How wise his daughter was, thought Naseeruddin, and congratulated himself on giving her a name worthy of her. Leonhard Eulers Solution of a Problem Belonging to the "Geometry of Position"  EMBED MSPhotoEd.3  1. Beside that part of geometry which deals with quantities and which always is studied with the greatest care, Leibniz makes mention of another part. He was the first to do so, although it is almost unknown, and he called it geometria sirus (geometry of position). This part of geometry was stated by him to deal only with the determination of location and with eliciting the properties of location. In this business no regard is had to quantities nor is there any need for calculating quantities. However, this does not sufficiently define the sort of problems that belong to this geometry of position, nor what method is to be used in solving them. Therefore, since recently there has been made mention of a certain problem, which seems to pertain to geometry, but which is so constituted that it requires neither quantitative determination nor admits of quantitative solution through calculation, I have not had any doubt at all to refer it to the geometry of position, especially because in its solution only position comes under consideration, while calculation is of no use. Hence I have determined here to exhibit my method, which I have invented for solving problems of this kind, as an example of the geometry of position. 2. The problem, then, which I was told is quite well enough known, was the following: In Knigsberg in Prussia there is an island A, called the Kneiphof, encircled by a river which divides into two arms, as can be seen from the figure: the branches are furnished with seven bridges, a, b, c, d, e, f, and g. Now the following question is asked concerning these bridges: Could someone follow a course so that he crosses each bridge once, and none more than once? I was told that some deny altogether that this is possible while others doubt it, but nobody asserts that it is possible. I formulated for myself the following general problem from this: whatever be the shape of the river and the distribution of its branches and whatever be the number of bridges, to find whether it is possible to cross all bridges once only, or not.  INCLUDEPICTURE "http://www.math10.com/en/maths-history/math-topology/imgFig1.jpg" \* MERGEFORMATINET  3. But since the problem of Knigsberg pertains to seven bridges, it could be solved by a complete enumeration of all the routes which can be taken; from this it would become clear whether some route satisfies the problem or not. But because of the great number of combinations this mode of solution is both too difficult and laborious and in other problems of more bridges cannot be employed at all. If this kind of method should be pursued to the end, many answers will be found to questions that were not asked; in this without doubt lies the cause of great difficulty. Wherefore having dismissed this method I have searched for another one which would not do anything more than show whether such a route can be found or not; for I suspected that such a method would be much simpler. 4. Now my whole method rests on a suitable way of designating each single crossing of the bridges; for this I use the capital letters A, B, C, D which describe each of the regions that are separated by the river. Thus, if someone goes from region A to region B, by either the bridge a, or the bridge b, I denote this crossing by the letters AB. The first of these shows the region from which the traveler came, and the second gives the region into which he goes after crossing the bridge. Again, if a traveler should go from region B to region D by bridge f, this crossing is represented by the letters BD. Two successive crossings AB and BD I then denote by the three letters ABD, because the middle letter B designates both the region which he reached by the first crossing and the region which he left by the second crossing. 5. Similarly, if the traveler should go on from region D to region C by the way of bridge g, I denote these three successive crossings by the four letters ABDC. From these four letters ABDC it will be understood that the traveler was fist in region A and crossed into region B, that from here he went on to region D and that from here he finally proceeded to C. Since, however, these regions are separated from each other by the river, it is necessary that the walker crossed three bridges. Thus crossings that are undertaken by way of four successive bridges are denoted by five letters; and if the walker crosses any number of bridges the number of letters denoting his route will be one greater than the number of bridges. Thus a crossing by seven bridges requires eight letters for its designation. 6. In this manner of denoting the crossings, I pay no attention to which bridges are used, but if the same crossing can be made from one region into another by several bridges, then it is just the same, whichever bridge is crossed, as long as the traveler reaches the designated region. From this it is clear that if the path over the seven bridges of the figure can be traced in such fashion that it crosses over each one once but over none twice, then this path can be represented by eight letters and these letters must be disposed in such fashion that the letters A and B occur directly next to each other twice, because there are two bridges a and b joining regions A and B; similarly, the two letters A and C also should occur twice in immediate succession in this series of eight letters; then the sequence of letters A and D should occur once, and similarly the sequence of letters B and D and C and D must occur once. 7. The question is reduced to this, then, that from the four letters A, B, C, and D a series of eight letters must be formed, in which all the sequences occur just as many times as we have indicated. However, before beginning work to find such an arrangement, it is convenient to show whether these letters can be disposed in this manner or not. For if it can be demonstrated that such an arrangement can by no means be made, all labor would be useless which was directed toward bringing this about. Wherefore I have searched for a rule, by means of which it could easily be ascertained-both for this question and for all similar ones-whether such an arrangement of letters can exist. 8. In order to find this rule I consider the single region A, into which any number of bridges a, b, c, d, etc. lead (Figure 7-2). Of all these bridges, I first pay attention to the single one a, which leads to the region A. If now the traveler crosses by way of this bridge, he necessarily must either have been in region A before he crosses, or must reach region A after the crossing. Therefore, according to the way of naming the crossing that I established above, it is necessary that the letter A occur once. If three bridges, say a, b, c, lead to region A and the traveler crosses over all three, then in naming his travel the letter A will occur twice, whether or not he started his course from A. Similarly, if five bridges lead to A, then in naming crossings by way of all five, the letter A must occur three times. And if the number of bridges be any odd number whatever, if we add one to this number and take half of it, this will give the number of times that letter A must occur. 9. To turn now to the case of the bridges which are to be crossed in Knigsberg. Because five bridges a, b, c, d, e lead to the island A, the letter A must occur three times in naming  INCLUDEPICTURE "http://www.math10.com/en/maths-history/math-topology/imgFig2.jpg" \* MERGEFORMATINET  the crossings over these bridges. Because three bridges lead to the region B, the letter B must occur twice, and similarly the letter D and the letter C must each occur twice. Hence in the series of eight letters, by which the crossing of seven bridges must be designated, the letter A should occur three times, and the letters B, C, and D each twice. But in a series of eight letters this can in no way be accomplished. From this it is clear that the required crossing over the seven bridges of Knigsberg cannot be done. 10. In similar fashion we can decide in any other case of bridges, if only the number which leads to any region is odd, whether each single bridge can be crossed just once. If it happens that the sum of all the times that each single letter should occur is equal to the number of all the bridges plus one, then such a crossing can be made. But if, as happened in our example, the sum of all the times should be greater than the number of bridges plus one, then such a crossing cannot be accomplished. The rule which I have given for finding the number of times of the letter A from the number of bridges leading into the region A, is equally valid whether all bridges come from one region B, as is the case in Figure 7-2, or whether they come from different regions; for I only consider the region A and inquire, how many times the letter A ought to occur. 11. If, however, the number of bridges leading to region A is even, then it must be known, in the matter of crossing each single bridge, whether the traveler began his course in A or not. For if two bridges lead to A and the traveler begins his course in A, then the letter A must occur twice; for it must once be present in order to denote the exit from A by one bridge, and once more in order to designate the reentry into A by way of the other bridge. But if the traveler begins his course in some other region, then the letter A will occur only once; for being written once it will denote both the arrival at A and the exit from A, in my manner of denoting such a course. 12. Now let four bridges lead into region A and let the traveler begin his course in A. In the designation of this course the letter A must be present three times, if he crosses over each single bridge once. But if he begins to walk in another region, then the letter A will occur only twice. If six bridges lead to the region A, then the letter A will occur four times, if the beginning of the walk is made at A; but if the traveler does not at the beginning come from A, then it will have to occur only three times. Generally therefore, if the number of bridges is even, one half of that number gives the number of times which the letter A must occur, if the beginning of the route is not in the region A; one half of the number of bridges plus one will give the number of times that the letter A must occur, if the beginning of the route is made in A itself. 13. Because in such a course the beginning can only be made in one region, I determine the number of times that the letter designating each region must occur from the number of bridges leading into the region, as half the sum of all the bridges plus one, if the number of bridges is odd; and as half of the number of bridges themselves, if it is even. Then, if the number of all the letter occurrences equals the number of the bridges plus one, the desired course can successfully be traversed; but the beginning must be made from a region into which an odd number of bridges leads. If, however, the number of letter occurrences should happen to be less by one than that of the bridges plus one, then the course can successfully be traversed by beginning in a region into which an even number of bridges leads, because in this way the number of letter occurrences is increased by one. 14. Suppose then that any configuration whatever of water and bridges is given and that it is to be investigated whether it is possible to cross over each bridge once; I go about it in the following fashion: First, I name all regions that are separated by water from each other by the letters A, B, C, etc. Second, I take the number of all the bridges, add one to it, and place this number at the head of the succeeding calculation. Third, after the letters A, B, C, etc., written below one another, I write the number of bridges leading into the region. Fourth, I mark with an asterisk those letters that have even numbers after them. Fifth, I write half of the even number next to each of the even numbers, and I write a number equal to half of each odd number plus one next to each odd number. Sixth, I add together the numbers written in the last column. If this sum is equal to, or less by one than the number of bridges plus one-then I conclude that the desired crossing can be made. But it must be noted that, if the sum is one less than the number placed above, then the beginning of the route must be made in a region marked with an asterisk; on the other hand, from a region not so marked, if the sum is equal to the number in question. Thus in the case of Knigsberg I make the following calculations: Number of bridges 7; Total number: 8 Bridges Crossings A 5 3 B 3 2 C 3 2 D 3 2 Because this calculation results in a sum greater than 8, a crossing of this kind cannot be made in any way. 15. Let there be two islands A and B, surrounded by water, and let this water be connected with four rivers, as the figure (Figure 7-3) shows. So that the island can be reached let there be 15 bridges a, b, c, d, etc. across the water surrounding the islands and the rivers. The question is whether some course can  INCLUDEPICTURE "http://www.math10.com/en/maths-history/math-topology/imgFig3.jpg" \* MERGEFORMATINET  be found so that each of the bridges is crossed, but none more than once. First, therefore, I name all the regions which are separated by water from one another, by the letters A, B, C,D, E, F; there are six of these regions. Then I add one to the number 15 of the bridges, and place the sum 16 at the head of the following calculation: Key number = 16 A* 8 4 B* 4 2 C* 4 2 D 3 2 E 5 3 F* 6 3 = 16 Third, I write the letters A, B, C, etc. under one another and with each I place the number of bridges that lead into this region, as 8 bridges lead to A, and four to B, etc. Fourth, those letters which have even numbers attached I mark with an asterisk. Fifth, in the third column I write half of the even numbers, but to the odd numbers I add one and write half of that. Sixth, I add the numbers of the third column to one another and obtain the sum 16. Since this is equal to the number 16 placed above the calculation, it follows that the crossing can be made in the desired fashion, if the course takes its beginnings either in region D or E, because these are not marked with an asterisk. The course could be made in this way: EaFbBcFdAeFfCgAhCiDkAmEnApBoElD, where I placed the bridges by which the crossings are made between the capital letters. 16. By this reasoning it will be easy to judge in every case no matter how greatly complex, whether all bridges can be crossed just once, or not. I shall now relate a much easier way of discerning the same thing, which follows without great difficulty from the present way, after I have first made the following observations. First I observe that all the numbers of bridges, written in the second column after the letters A, B, C, etc. if added together are twice as great as the number of bridges. The reason of this is that in this calculation where all bridges leading into a given region are counted, each bridge is counted twice; for each bridge has reference to both regions which it joins. 17. From this observation it follows therefore that the sum of all the bridges which lead into each region is an even number, because its half is equal to the number of bridges. Hence it cannot happen that among the numbers of bridges leading into the several regions there is just one that is uneven; nor that three be uneven, nor five, etc. Hence if any of the numbers signifying the bridges, attached to the letters A, B, C, etc. are uneven, it is necessary that the number of these numbers be even. Thus in the example of Knigsberg there were four numbers of bridges that were odd, attached to the letters of the regions A, B, C, D, as can be seen from section 14. And in the preceding example, in section 15, there are only two odd numbers, attached to the letters D and E. 18. Since the sum of all the numbers attached to the letters A, B, C, etc. equals twice the number of bridges, it is apparent that if two be added to this sum and the result divided by 2, then this must give the number placed at the head of the calculation. If, therefore, all the numbers attached to the letters A, B, C, D, etc. are even and in order to obtain the numbers of the third column half of each of them is taken, their sum will be less by one than the key number at the top. Therefore in such cases a crossing over the bridges can always be made. For in whatever region the course begins, it has bridges even in number leading to it, as is required. Thus in the Knigsberg case it would be possible for someone to cross over each bridge twice; each bridge could be, as it were, divided in two, and then the number of bridges leading into each region will be even. 19. Furthermore, if only two of the numbers attached to the letters A, B, C, etc. are odd, but all the others are even, then the desired crossing can always be successfully made, as long as the beginning of the course is in a region with which an odd number of bridges connect. For if the even numbers are halved as well as the odd numbers plus one, according to the rule, the sum of all these halves will be greater by one than the number of bridges and therefore equal to the key number at the head. From this it will then be seen that, if there are four or six or eight, etc. odd numbers in the second column, then the sum of the numbers in the third column will be greater than the key number at the head and will exceed it by one or two or three etc. and hence the crossing cannot be made. 20. Hence if any case whatsoever be given, it can now very easily be recognized whether a crossing over all bridges once can be made or not, with the help of this rule: If there are more than two regions which have an odd number of bridges leading to them, then it can with certainty be affirmed that such a crossing cannot be made. If, however, there are two regions which have an odd number of bridges leading to them, then the crossing can be made, if the course begins in one of these regions. If, finally, there are no regions which have odd numbers of bridges leading to them, then the desired crossing can be made, no matter in which region the beginning of the walk is made. This rule therefore fully solves the given problem. 21. But when it has been found that such a crossing can be made, the question still remains, how the course is to be found. For this I use the following rule: Let pairs of bridges which lead from one region to another, be eliminated in thought, as many times as it can be done. In this way, the number of bridges will be radically and quickly diminished. Then the desired course over the remaining bridges which can easily be done is looked for. When this has been found, it will at once be clear to anyone who attends to it that the bridges eliminated in thought will not disturb this course: and I judge it is not necessary for me to teach more about the finding of the course. 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