Galileo and the Magic Numbers
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by Sidney Rosen
Description: Sixteenth century Italy produced a genius who marked the world with his studies and hypotheses about mathematical, physical and astronomical truths. His father, musician Vincenzio Galilei said, "Truth is not found behind a man's reputation. Truth appears only when the answers to questions are searched out by a free mind. This is not the easy path in life but it is the most rewarding." Galileo challenged divine law and the physics of Aristotle, and questioned everything in search of truths. And it was on this quest for truth that he was able to establish a structure for modern science.
eBook Publisher: E-Reads, 1958
eBookwise Release Date: August 2001
9 Reader Ratings:
Available eBook Formats: OEBFF Format (IMP) [233 KB]
Reading time: 152-213 min.
I GALILEO lay on his back, hands under his head, and stared up at the crack that zigzagged across the ceiling. There was just enough moonlight coming through the bedroom window to follow the dark line in the plaster. Outside, all about his house, the city of Pisa slept peacefully. Somewhere far away, Galileo could hear the faint clumping of horses' feet on cobblestones. It was probably the night watch riding through the streets.
On the other side of the room, his younger brother, Michelangelo, turned restlessly in his bed and muttered, "Good--good doggie." He must be dreaming of the puppy Father promised him, Galileo thought. His sister, Virginia, was still a baby and slept in their parents' bedroom.
The weather was cold--it was the middle of February--and the boys' bedroom was chilled. But Galileo was too excited thinking about the next day to notice that his blanket had fallen to one side.
Tomorrow, he said to himself, I will be nine years old. Nine years, that's a long time to have been alive. And tomorrow I begin my studies at the school of Master Borghini. I wonder what that will be like. Will he beat me if I do not know my lessons?
He remembered what his father had said about school the day before. Galileo had asked him, "Why must I go away to school. Father? Why can't you teach me more here at home?"
His father had shaken his head. "There are certain things I can teach well, son, and certain things I can't teach at all. Remember, going to school is a privilege--the privilege of nobility. The sons of poor commoners have to go to work by the time they are your age--there's no school for them. But for you, the son of Vincenzio Galilei, musician at the Court of Florence, there will be schooling. Be glad you are one who can go to school!"
And Galileo, who loved his father very much and would do anything to please him, clapped his hands and cried, "I am glad, Father, I am!"
Now Galileo began to practice his Latin declensions, conjugations, and grammar rules out loud, so that he would be ready for the next day.
"Canto, I sing; cantos, you sing; cantat, he sings. Sum, I am; fui, I was. Qui, quae, quod, cuius, cuius, cuius, cui, cui--"
"What is all this commotion in here!" His mother's voice hissed sharply as the door was flung open.
"I was just practicing my Latin for tomorrow. Mother."
"Oh, all that nonsense about learning and education! Better your father got some sense into his head and apprenticed you to some rich merchant! But these Galileis with their notions about nobility and learning! Now, be quiet, or you'll wake your brother, and then I'll have a job on my hands. You may be nine years old, young man, but you're not old enough so that I can't give you a good spanking! Now, go to sleep!"
"Yes, Mother." Why did Madam Giulia always have to be shouting at him, he wondered. I'll practice quietly, he decided. Ab, ante, con, in, inter, ob, post, prae ... suddenly the words seemed far, far away. A moment later, Galileo was fast asleep.
In the morning, his father came into the room while he was dressing. Michelangelo was in the kitchen having breakfast.
"Inspection!" cried Vincenzio gaily. "Turn around! Point one--hose and breeches clean. Right! Point two--hair trimmed and combed. Right! Point three--jerkin clean and no wrinkles. Right! Point four--shoes brushed. No, not right!"
He pointed to a large spot of dirt on Galileo's right shoe.
"Sorry, Father." Galileo hastened to wipe at the thick-soled leather shoe with a cloth.
"Remember, you are Galileo Galilei, son of Vincenzio Galilei. We are a noble family. Remember that one of your ancestors was Tomaso Galilei, one of the Twelve Good Men of Florence. There are certain responsibilities that go with noble breeding: cleanliness, learning, good manners."
Galileo had heard this little speech many times in the last few years. But he loved his father very much. For Galileo, Vincenzio represented all that was beautiful and kind and noble in the world. He had taught Galileo the rudiments of Latin and Greek. And from him, Galileo had learned to do one of the things he loved best--to play the lute. After all, when a fellow's father is a music teacher--and good enough to have to make a special trip every week to teach at the Grand Duke's court in Florence!--the least a fellow can do is learn to play music well.
This morning, his father had brought his lute with him into the bedroom. "Well, just to cheer you up before you go to school, I'll teach you a new ballad."
Galileo took up his own lute from the corner and made ready to follow his father's fingering on the strings. One of the middle strings was slightly out of tune. He twisted one of the tuning pegs until the string struck just the right note.
"This is a noble ballad in the French tongue," said Vincenzio. "I will sing it that way first, and then I will translate for you."
L'homme, l'homme, l'homme armé,
L'homme armé, l'homme armé doibt on doubter, doibt on doubter,
On a fait partout crier:
Que chascun se veigne armer
D'un haubregon de fer!
"That's a very fetching tune. It makes me feel like marching. What do the words mean?"
"This ballad is a humorous one, describing how the people feel about knights in armor."
Oh, the man, the man, the man at arms,
He fills us all with dread alarms;
Everywhere, the people wail:
Find, if you would breast the gale,
A good stout coat of mail!
Galileo and Vincenzio played the music over together, and then sang the whole ballad through once. Not a mistake. Vincenzio was very proud of his son.
"Are you two lazy fellows going to spend the whole day in that bedroom? The breakfast is cold. It is almost eight o'clock. That son of yours will be late for his precious lessons, Vincenzio!"
Why does she have to scold so? Galileo felt annoyed with his mother. Her fit of bad temper was nothing new. As far back as he could remember, Galileo had heard his mother's voice raised in shrill argument. She screamed at Vincenzio about the lack of money in the house. She nagged at the children day and night about a thousand different little things.
Why couldn't I have a mother who is kind and pleasant like my father? thought Galileo. But then he suddenly remembered a night a few months before, when he had lain in his bed burning and tossing with fever. His mother had sat beside him all night, not sleeping, soothing and cooling him as best she could. His scolding mother could be a kind and loving angel at the right moments.
Galileo sighed. He and Vincenzio went sheepishly to eat their cold breakfast.
It was time to leave for Signor Borghini's house. Galileo excused himself from the table and went for his woolen coat, a hand-me-down that was almost threadbare. The coat was gone! Galileo ran back to the dining room.
"Something wrong?" asked his father.
"My coat--my coat--it's--it's--"
Michelangelo mocked him: "It's--it's--"
Vincenzio laughed. "Oh, that old thing! I gave that to a beggar yesterday."
"Then I cannot go to school?"
"Look in the corner there."
Galileo turned. Hanging on a hook on the wall was the most beautiful winter cape he had ever seen, made in the Spanish style. The collar was cut square and stood partly erect; its edge was trimmed with soft fur.
"Happy birthday, Galileo!" cried Vincenzio and Madam Giulia and Michelangelo. Even the baby, Virginia, though she could not yet speak, gurgled and clapped her hands to show that she wanted to play the game too.
Vincenzio placed the cloak about Galileo's shoulders. It fitted perfectly. "Well, look at our handsome little nobleman now!"
"I want a birthday present, too!" cried Michelangelo. Vincenzio tried to explain to him that he would get a present when his birthday came, but to no avail. Michelangelo cried and fussed and had to be sent out of the room.
"I must admit Galileo looks elegant," sighed Madam Giulia, "but it's such an expensive coat."
"A present from a nobleman at the Florentine Court. I gave his son a lute lesson or two. Naturally, he was grateful."
"Naturally, I would have been more grateful for a few gold florins for food and better clothing for all of us!"
"Ah, stop nagging, woman!" shouted Vincenzio.
"Why don't you find a more generous patron, then? This one squeezes a florin ten times before it leaves his fingers!"
Galileo was embarrassed by the argument. He tried to change the subject. "What does Mother mean by a patron?" he asked his father.
Vincenzio appeared a little abashed. "Well, a patron is like a--well, like a protector. In these times, painters and musicians and writers look to some wealthy nobleman to support them while they create works of art."
"Just a fancy way of begging!" put in Madam Giulia.
"That is not true! It's just the way of the times, that is all. Otherwise, artists would starve, and there would be no great paintings, no beautiful music, no magnificent statues, no books. You understand, Galileo?"
"Yes, Father, I understand."
"There is no shame connected with patronage, as your mother would have you believe. Well, that's enough of that. Here are your books. Off with you, my boy!"
Galileo, pretending he was a grand duke, with head and nose high in the air, minced past his parents and out the front door. His father roared with laughter, and even his mother couldn't help giggling. For a nine-year-old, their son had a fine sense of humor.
Master Jacopo Borghini was a small dried-up man, upon whose thin and wrinkled face sat immeasurable sadness. He took Galileo's new cloak and gazed at it a moment before he hung it up.
"Noble clothing, eh? Well, I'm more concerned with new ideas in the head than I am with new clothes on the back. Now, tell me: how far have you gone in your Latin?"
For the next thirty minutes, Borghini quizzed Galileo closely on Latin forms and grammar. The Latin went well. Next came Greek. This didn't take as long, because Vincenzio's understanding of Greek had been rather poor.
"Well, we'll have to work on that Greek grammar. Now, how about mathematics?"
"Mathematics? You mean arithmetic? I can add and subtract numbers."
"Do you know Pythagorean number magic?"
Galileo shook his head. He knew about witches and black magic, but number magic--? That was a new kind of magic.
Borghini went to the cupboard and returned with a handful of little white pebbles.
"Pythagoras was a Greek philosopher who lived over two thousand years ago. He loved numbers. For him, the whole universe could be explained by mathematics. He thought numbers could describe beauty, music, and even the acts of gods and men. Your father is a musician, is he not?"
"Yes, Master, and a fine one."
"So I am told. Then you will understand what I mean when I say that Pythagoras invented the first numbered musical scale."
"He must have been a very great man, indeed, to have done that."
Master Jacopo knelt on the floor and motioned Galileo to do the same. Galileo's eyes widened. This was a strange way for a teacher to act! Most of his friends had told hen dreadful stories of their teachers. They all insisted upon strict discipline and were very formal and stiff. Yet this teacher was asking him to sit on the floor to play games! Galileo sat, legs crossed like a Saracen.
'Now, this is the magic," said Borghini. He placed one pebble on the floor.
'What number is that?"
"Now I make a triangle by adding two pebbles. A triangle is a figure with three sides. Observe."
"What number now?"
Master Jacopo added three pebbles. The triangle grew larger.
"That makes six!"
Now the triangle became very large, as four pebbles were added.
"That makes ten!"
"Correct. Now, what were the first three triangles?"
"I remember. One and three and six." Galileo's eyebrows rose in excitement. "But that adds up to ten!"
"That is the magic. Master Galileo. The first three triangles add up to make the fourth one. And each time, we added only one more pebble than the previous time!"
For some reason he could not understand, Galileo thought this was the most wonderful thing he had ever learned! And when Master Jacopo arose, dusted off his hose, pat away the pebbles, and said, "Now for those Greek conjugations," Galileo was reluctant to move.
But when Borghini added, "Perhaps tomorrow will bring another kind of number magic," Galileo leaped to his feet. He was ready to struggle with dull Latin or Greek--as long as there was promise of more Pythagorean games.
Almost a year and a half passed. Galileo came home one day to find the house in an uproar. Everything had been taken off the pantry shelves, and all the closets had been emptied. Clothing was strewn over tables and chairs. In the center of the kitchen, Madam Giulia was packing dishes into a large cask; with the other hand, she fended off little Virginia, who kept getting in her way. Madam Giulia's face was red from exertion, and her breath came in gasps.
"Get away--you pestiferous child--go away and play--a plague on this packing--Michelangelo! Michelangelo!"
Galileo's brother came running in from the bedroom.
"Take your sister out of doors and play with her. Do something, anything, but get her out of my way!"
Michelangelo picked up Virginia. The little girl laughed happily at what she thought was a wonderful game. Michelangelo, laughing and teasing, carried her outside.
"Are we moving away. Mother?" asked Galileo.
"No, of course not, this is a game I'm playing," scolded his mother. Then she looked up and saw the surprise on Galileo's face. "Glory to heaven, didn't you know? Your father again! He forgot to tell you this morning. We're moving to Florence!"
"To Florence!" Galileo's eyes brightened. What an adventure! Florence was the center of Tuscany, where the leading noble family, the Medici, lived in their marble palaces. Florence was the great city, full of wonderful people, wonderful streets, wonderful things! In Florence, there lived famous painters and sculptors. Galileo had heard of one, a man named Leonardo da Vinci. This man had invented huge machines of war and had painted many beautiful pictures. It was said of him that he had made a machine that would enable a man to fly like a bird!
Galileo loved to make little toys that moved of themselves. He made, of sticks and little flat stones, a windmill that moved and really ground little kernels of wheat into flour. He made a sailboat that steered itself across a rain-water puddle. But a machine that had wings and flew like a bird! That was the invention of a great master!
"Well, don't stand there like an idiot! Get a box and begin to pack your things!"
"Why are we going to Florence?" asked Galileo, starting from his dream about Leonardo.
"You know your father teaches music at the court! Now it is to be a full-time position, and no more week-end traveling! The pay isn't much, but at least it will be steady. So, we are moving into a house in Florence. Now, hurry; we leave the first thing in the morning!"
Galileo made ready the little clothing he had and the few possessions that were his very own--his books, his lute, the little windmill that ground wheat, a drawing he had made of the Leaning Tower, and a small bag full of round pebbles. These last had become most precious, for with them he had practiced the number magic of Pythagoras.
"Well, boy, don't just sit there staring into space. Get a move on with your packing!" Galileo's father came into the room. "Are you glad to be going to Florence?"
"Yes, Father. You know, every day, as I crossed over the Arno River to go to school, I used to look up the river to the east and think: there is the great city of Florence, where my father teaches noble lords and ladies to play the lute. Some day, I shall go there myself and see all the great palaces and the noble people. And now we are going, and I shall see them!"
Vincenzio ruffled his son's hair. "Not if you don't hustle with that packing. And when you're through, give your mother a hand, or she'll have our hide!" He winked conspiratorially at Galileo.
At sunrise the next morning, a cart stood before the door. The day promised to be cloudless and warm--a good omen. Galileo awoke to the sound of men's voices.
"I think ten florins is not too high for such a load!"
"Ten florins! Man, what are you thinking of? This isn't the Grand Duke of Tuscany you're moving. We'll give you six!"
Galileo recognized his uncle, Muzio Tedaldi, who had promised to come and help with the moving. The other voice apparently belonged to the wagoner who was to drive them to Florence.
Six! It's hardly worth it!"
"Muzio! Guido! What's all this wrangling?" He heard his father intervene. "Let's settle it for eight florins. Agreed?"
"Too much!" cried Muzio. But the wagoner muttered, "Agreed!" A moment later, the three men were laughing together at a joke Vincenzio had made. How wonderful it was to have a father who could settle disputes so wisely!
Galileo leaped out of bed, awakened Michelangelo, and threw his clothing on in a frenzy. He felt almost delirious with excitement and joy.
The wagon was soon filled with the meager possessions of the Galilei family, and the simple breakfast was quickly eaten. Vincenzio helped Madam Giulia up into the wagon, and then tossed Virginia up to her. He lifted Michelangelo and swung him up beside his mother.
"All present and accounted for? Then off we go!
The wagoner cracked his whip, the horse strained at the harness, and away they went. Galileo sat facing Pisa and waving good-by to his uncle. Soon, Signor Tedaldi was only a little speck, almost hidden by the dust of the road. Galileo turned around.
Out there before them, the road ran eastward all the way to the great, splendid city of Florence. Vincenzio had already taken up his lute and was singing.
L'homme, L'homme, L'homme armé, L'homme armé,
L'homme armé doibt on doubter--
As loudly as he could, Galileo joined in. Their voices rang out over the rumbling of the wheels, as the little wagon rattled along the dusty road to Florence.