ARCHIMEDES: The Works of Archimedes

 ARCHIMEDES, THE WORKS OF ARCHIMEDES

QUOTES FOR DISCUSSION

BIOGRAPHICAL NOTES:

Archimedes 287-212 BC

He may have studied with the pupils of Euclid in Alexandra.

Archimedes, Biographical note, Great Books Volume 11,  pg 399

After discovering the solution of the problem, “To move a given weight by a given force,” he boasted to King Hiero: “Give me a place to stand on and I can move the earth.” Asked for a practical demonstration, he contrived a machine by which with the use of only one arm he drew out of the dock a large ship, laden with passengers and good, which the combined strength of the Syracusans could scarcely move.

Archimedes, Biographical note, Great Books Volume 11, pg 399

Unlike Euclid and Apololonius he wrote no textbooks. Of his writings, although some have been lost, the most important have survived.

Archimedes, Biographical note, Great Books Volume 11, pg 399

The absorption of Archimedes in his mathematical investigations was so great that he forgot his food and neglected his person, and when carried by force to the bath, Plutarch records, “the used to trace geometrical figures in the ashes of the fire and diagrams in the oil on his body.” Asked by Hiero to discover whether a goldsmith had alloyed with silver the gold of his crown, Archimedes found the answer while bathing by considering the water displaced by his body, whereupon he is reported to have run home in his excitement without his clothes, shouting, “Eureka,” (I have found it).

Archimedes, Biographical note, Great Books Volume 11, Pg 400

In accordance with the expressed desire of Archimedes, his family and friends inscribed on his tomb the figure of his favorite theorem, on the sphere and the circumscribed cylinder, and the ratio of the containing solid to the contained.

Archimedes, Biographical note, Great Books Volume 11, Pg 400

ARCHIMEDES ON THE SPHERE AND CYLINDER

Archimedes to Dositheus greeting:

“On a former occasion I sent you the investigations which I had up to that time completed, including the proofs,…”

Archimedes, The Works of Archimedes: On the Sphere and Cylinder, Book One, Great Books Volume 11, Pg 403

Now, however, it will be open to those who possess the requisite ability to examine these discoveries of mine.

Archimedes, The Works of Archimedes: On the Sphere and Cylinder, Book One, Great Books Volume 11, Pg 403

The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.

Archimedes, The Works of Archimedes:  On the Sphere and Cylinder, Book One, Measurement of a Circle, Great Books Volume 11,  Pg 447

Therefore the circumference of the circle(being less than the perimeter of the polygon) is a fortiori less than 3 1/7 times the diameter AB.

Archimedes, The Works of Archimedes: On the Sphere and Cylinder, Book One, Measurement of a Circle, Great Books Volume 11, Pg 450

ARCHIMEDES ON THE EQUILIBRIUM OF PLANES OR THE CENTRES OF GRAVITY OF PLANES

Proposition 1

Weights which balance at equal distances are equal. For, if they are unequal, take away from the greater the difference between the two. The remainders will then not balance {Post.3}; which is absurd. Therefore the weights cannot be unequal.

Archimedes , The Works of Archimedes:  On the Equilibrium of Planes or the Centres of Gravity of Planes Book 1, Great Books Volume 11, pg 502

ARCHEMIDES: THE SAND-RECKONER

There are some, King Gelon, who think that the number of the sand is infinite in multitude; and I mean by the same not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. … But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the word which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the earth filled up in the way described, but also that of a mass equal in magnitude to the universe.

Archimedes, The Works of Archimedes: The Sand-Reckoner, Great Books Volume 11, pg 520

Now you are aware that ‘universe’ is the name given by most astronomers to the sphere whose centre is the centre of the earth and whose radius is equal to the straight line between the centre of the sun and the centre of the earth.

Archimedes, The Works of Archimedes: The Sand-Reckoner, Great Books Volume 11, pg 520

2. “The diameter of the earth is greater than the diameter of the moon, and the diameter of the sun is greater than the diameter of the earth.”

Archimedes, The Works of Archimedes: The Sand-Reckoner, Great Books Volume 11, Pg 521

Orders and periods of Numbers

I. We have traditional names for numbers up to a myriad (10,000); we can therefore express numbers up to a myriad myriads (100, 000,000). Let these numbers be called numbers of the first order.

Suppose the 100,000,000 to be the unit of the second order, and let the second order consist of the numbers from that unit up to (100,000,000)2 .

Let this again be the unit of the third order of numbers ending with (100,000,000)3 ; and so on, until we reach the 100,000,000th order of numbers ending with (100,000,000) 100,000,000 , which we will call P.

II. Supposed the numbers from 1 to P just described to form the first period. Let P be the unit of the first order of the second period, and let this consist of the numbers from P up to 100,000,000P.

Let the last number be the unit of the second order of the second period, and let this end with (100,000,000)2 P, or P2

We can go on this way till we reach the 100,00,000th order of the second period ending with (100,000,000) 100,000,000 P, or P2

Archimedes, The Works of Archimedes: The Sand-Reckoner, Great Books Volume 11, pg 524

… This last number is expressed by Archimedes as “a myriad-myriad units of the myriad-myriad-th order of the myriad-myraid-th period which is easily seen to be 100,000,000 times the product of (100,000,000) 99,999,999 and P99,999,999, i.e. P100,000,000

Archimedes, The Works of Archimedes: The Sand-Reckoner, Great Books Volume 11, pg 524

Hence the number of grains of sand which could be contained in a sphere of the size of our “universe” is less than 1,000 units of the seventh order of numbers or 1051

Archimedes, The Works of Archimedes: The Sand-Reckoner, Great Books Volume 11, pg 526

Conclusion:

“I conceive that these things, King Gelon, will appear incredible to the great majority of people who have not studied mathematics, but that to those who are conversant therewith and have given thought to the question of the distances and sizes of the earth, the sun and moon and the whole universe, the proof will carry conviction. And it was for this reason that I thought the subject would be not inappropriate for your consideration.”

Archimedes, The Works of Archimedes:  The Sand-Reckoner, Great Books Volume 11, pg 526

ARCHIMEDES ON FLOATING BODIES

Proposition 6

If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the difference between its weight and the weight of the fluid displaced.

Archimedes, The Works of Archimedes: On Floating Bodies, Book One, Great Books Volume 11, Pg 540

Postulate 2

“Let it be granted that bodies which are forces upwards in a fluid are forced upwards along the perpendicular (to the surface) which passes through their centre of gravity.”

Archimedes, The Works of Archimedes: On Floating Bodies, Book One, Great Books Volume 11, Pg 541

BUCK, Pearl S. The Pavilion of Women

PEARL BUCK: THE PAVILION OF WOMEN

QUOTES FOR DISCUSSION

Since he never had a daughter, he had put aside the convention that forbade a man to speak to his son’s wife. Many times he had even sent for her so that he might read to her something from the books in his library. It gave her pleasure to think that library full of books was now hers. Today, after years of giving her body and mind to others, he she felt she needed to drink deeply at old springs.
- pg 20

Madame Wu slept all that night without waking. When she awoke in the morning, she was completely rested. Fatigue had left her body. But there was something familiar in this feeling. Thus she had felt after each of her children had been born. Her first thought when she heard that sharp cry of the new child was always of reclaiming her own freedom. That joy of freedom was in her again.
- pg 40

 “They want to look at you,” she explained to brother André

Why not?” he replied and turned himself toward them.

The children shrank back at this, but when he remained motionless and smiling, they came near again.

“Why are you so big?” a child asked breathlessly.

“God made me so,” Brother André replied.
- pg 50

 “You are very lonely,” she said abruptly. “All day you work among the poor and at night among the stars.”

“It is true,” he agreed calmly.
- pg 72

 “Are there other men like you?” she asked.

“No man is quite like any other one, “ Brother André said.

His sun-browned face took on a warm, almost smiling look. “But your son, young Fengmo, I think he could become like me. Perhaps he will become like me.”

“I forbid it!” Madame Wu said imperiously.

“Ah!” Brother André said, and now he smiled. His eyes glowed for an instant, and then he said good bye. And she sat gazing up into the handful of stars above her court.
- pg 72

I thought if I did my duty to everyone, I could be free.”

“What do you mean by freedom?” He inquired.

“Very little,” she said humbly. “Simply to be mistress of my own person and my own time.”

“You ask a great deal,” he replied. “You ask for everything.”
- pg 93

 “Forget your own self,” he said.

“But all these years,” she urged, “I have so carefully fulfilled my duty.”

“Always with the thought of your own freedom,” He said.

She could not deny it.

“Instead of your own freedom, think how you can free others,” he said gently.

She lifted her head.

“From yourself,” he said still more gently.
- Pg 94

She had seen freedom hanging like a peach upon a tree. She had nurtured the tree, and when it bore, she had seized upon the fruit and found it green.
- Pg 94

 “Is our Chinese Heaven your God, and is your God our Heaven?” She inquired of Brother André.

“They are one and the same,” He replied.

“Then anywhere upon the round earth, by whatever seas, those who believe in any God believe in the one?” She asked.

“And so are brothers,” he said, agreeing.

“And if I do not believe in any?” she inquired willfully.

“God is patient,” he said. “God waits. Is there not eternity?”

She felt a strange warm current pass through him and through her. But it did not begin in him, and it did not end in her. They seemed only to transmit it, from the ends of the earth to the ends of the earth.

“Heaven is patient,” she repeated. “Heaven waits.”
- pg 97

Madame Wu closed the door. The foreign priest was neither foreign nor a priest to her now. She had never thought of him as a man when he was alive, but now that he was dead she saw him as a man lying dead. In his youth he must have been extremely handsome. The great body lying outstretched before her was of heroic proportions. His skin was pale and now in death was becoming translucent. Suddenly she recognized him. “You whom I love!” she murmured in profound astonishment. In the instant she accepted this recognition she felt her whole being change.
- pg 98

 “What is in that black box?” she asked and pointed

“That is a magic voice box,” the old woman said. “He used to listen to the voices in the night.”

Madame Wu remembered that he had told her of it. She approached the box and put her ear against it and heard nothing.

“It speaks for on one else,” the old woman explained.

“Ah, then we will bury it with him,” Madame Wu said.

There is one more thing he possesses, and it is magic, too,” The old woman said hesitantly. She then crawled under the bed and drew out a long wooden box. She opened it, and there lay an instrument, a pipe of some sort. “He held it to his right eye whenever the night was clear, and he looked into Heaven,” she said.
- pg 99

She sat awake and alone for hours that night, searching out the whole of her new knowledge. She loved a dead stranger, a man who had never once put out his hand to touch hers, whose touch would have been unthinkable. Had he lived, they would have accepted renunciation.
- pg 100

The next morning instead of waking to weariness and longing not to begin the day, she was aware of fresh energy in herself. What she felt now for André warmed and strengthened her. Love permeated her brain as well as her body. André was not dead. He was living, and he was with her because she loved him.
- pg 106

 “…it is a very grave thing to enter a large and honorable family such as ours. You can come into it and ruin all our happiness here. Or you can come in and add happiness by your presence.” - Pg 107

If Jasmine really loved Mr. Wu, that love, too, must be allowed. All the unhappiness in homes came because there was not love.
- Pg 108

The children looked at her with love, and suddenly for the first time in her life Madame Wu felt the true pangs of birth.
- Pg 109

He did make us laugh every day.
- Pg 109

Madame Wu did not understand fully the change that had taken place in her being. Indeed, she did not know from one moment to the net where her path lay. But she felt that she was walking along a path of light. And the light that lit this path was her love for André. When she needed to know what step should be taken next, she had only to think of him.
-  Pg 115

Madame Wu wiped her eyes delicately. While Tsemo was alive she had not missed him much, but now she missed him very much and thought of him often. She knew that what she missed was not what she had known, but what she had never known. She reproached herself very much that she had allowed a son to grow up in her house and had never really become acquainted with his being.
-  Pg 132

 “I have learned as I have grown older,” she said, “ There is a debt due to every soul, and that is the right to its own true happiness.”

“That is what Brother André use to say,” Fengmo said suddenly. Mother and son felt themselves drawn together, as though by some power or presence they did not see.
- Pg 132

 “He told me that in his village no one could read or write, and they had to go to the city to find a scholar. I had never understood the pity of this until I came to know him. He was very intelligent, but the poor old man could not read. Then I remembered that this is true in our villages, too. None of our own people can read and write either.

Why should they?” Madame Wu inquired. “They do not come and go. They only till the fields.”

“But Mother, “ Fengmo exclaimed, “to know how to read is to light a lamp in the mind, to release the soul from prison, to open a gate to the universe.”
- Pg 137

 “To lie is a sin, “ Brother Andé had taught him simply, “but it is not a sin against God so much as a sin against yourself.”
-  Pg 139

Even some of the old farmers wanted to learn when they saw how the younger ones profited by it, and Fengmo lost no chance to make it widely known when a young farmer gained by his ability to read a bill or check an account. Other villages asked for schools,
- Pg 140

As they reached the village Madame Wu was amazed at the changes she saw. It was clean and prosperous as it had never been. The children were clean and their hair brushed.
- Pg 143

 “I shall never leave our gates except to visit my mother.”
- Pg 145

 “No one in our country who has learning ought to keep it for himself,” Fengmo insisted.
- Pg 145

 “The soul of every creature must take its own shape, and no one can compel another without hurting himself.”
- Pg 148