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