Age of the Earth and Archimedes

        Age of the Earth  

          Although ancient rocks exceeding 3.5 Ga or giga-anni (billion years in age) are present on all continents, the oldest rock formations identified to date are the Acasta River gneiss, a granitoid-greenstone complex (dated around 4.03 to 4.06 Ga) in North West Territory, Canada, southeast of Great Slave Lake; and the Isua Supracrustal Formation in West Greenland (dated at 3.7 to 3.87 Ga). Other well-documented rocks nearly as old have been identified in the Minnesota River Valley and northern Michigan (3.5-3.7 Ga), the Pongola Supergroup in the Buffalo River gorge of South Africa and Swaziland (3.4-3.5 Ga), and in Western Australia (3.4-3.6 Ga). The ages of those rocks have been assessed by various radiometric dating methods; the consistency of the results provides scientific confidence that the ages are correct to within a few percentage points. An interesting feature of those ancient rocks is that they are not types of primordial crust but are lava flows and sediments that were deposited in shallow water, an indication that Earth history began well before those rocks were formed.
          In Western Australia, detrital and xenocrystic zircon crystals with U-Pb ages significantly older than 4 Ga have been found at five localities within the Yilgarn Craton in younger sedimentary rocks and have radiometric ages of about 4.35 to 4.4 Ga, making these tiny crystals the oldest materials to be found on Earth. The oldest specimen is from Jack Hills; its age of 4.404 Ga make it the most ancient fragment of the Earth’s crust that has been identified, formed approximately 150 million years after the planet itself, perhaps in an environment that contained liquid water and had considerably lower temperature than had been anticipated previously. That recent determination has profound implications in terms of the critical question as to when the Earth became habitable for life. Virtually all geoscientists agree that liquid water at or near the Earth’s surface was a prerequisite for the establishment of life. Consequently, if those environmental conditions are found by ongoing research to be as proposed, the Earth may have hosted life for as much as 700 million years longer than is currently believed. The best age for the Earth (4.54 Ga) is based on old, presumed single-stage leads coupled with the Pb ratios in troilite from iron meteorites, specifically the fragments found at Meteor Crater, Arizona.

          Author’s Note: Well, unless you’re firmly in the creationist camp, skipped over the above material, and have closed your mind to reason and rationality, the age of our Earth as established by various geosciences is approximately 4.6 billion years. But grasping the concept of the age of our planet really is not an easy task. Since thinking in conceptual terms about that ferociously long a time period is relatively fruitless and frustrating for most people, allow me to provide a more concrete example. Imagine those 4.6 billion years being squeezed down into a single year. In such a calendar, the time during which the first life-forms evolved (the Precambrian) would start January 1 and extend to about November 15, constituting more than five-sixths of all Earth history. No joke. For most of us, that’s quite a shock to our anthropocentric point of view. The age of invertebrates and primitive fishes, the Paleozoic, which extended from about 542 to 251 mya, would occupy the rest of November and part of December, or one-twelfth of the Earth’s age. The great age of the dinosaurs, the Mesozoic, which was from about 251 to 65 mya, would constitute most of the rest of December. The Quaternary, consisting of the Pleistocene and Holocene epochs or slightly less than two million years total, which would be equivalent to the time humans have spent on the Earth, would occupy only the last four hours on New Year’s Eve.

Archimedes[1]

          Famous Greek mathematician, 287 to 212 BCE, whose wide-ranging mind and pioneering ideas explored and expanded the worlds of physics, mechanics, and hydrostatics. He was famous for creating numerous geometric proofs using the rigid geometric formalism of Euclid, excelling especially at computing areas and volumes of spheres, cylinders, circles, and parabolas and was the first mathematician to correctly calculate the value of π.

          Historical Background: Few facts of Archimedes’s life are known with certainty but tradition has made at least two vignettes famous. In one story, he was asked by King Hiero II to determine whether a crown was pure gold or was alloyed with silver. Archimedes was perplexed, until one day, supposedly observing the overflow of water as he stepped into his bath, he suddenly realized that since gold is more dense (i.e., has more weight per volume) than silver, a given weight of gold represents a smaller volume than an equal weight of silver and that that given weight of gold would therefore displace less water than an equal weight of silver. Delighted at his discovery, he ran home sans clothes, shouting (perhaps apocryphally) “Eureka,” which means, “I’ve found it.” Let’s hope he was referring to the density theory rather than something more personal. When he showed that Hiero’s crown displaced more water than an equal weight of gold he proved the crown had been alloyed with another metal less dense than gold. We can only guess the jeweler’s unpleasant fate.

          In the other story Archimedes is said to have told Hiero II, illustrating the principle of the lever, “Give me a place to stand and I will move the world.” He invented extremely successful machines of war (in the Second Punic War in which Syracuse was allied with Carthage against Rome) so ingenious that Syracuse was able to hold off the besieging armies of Marcus Claudius Marcellus for three years (who was not only the commander of the Roman army but also the nephew of Caesar Augustus and his first designated heir, only to die before Augustus, plausibly from poisoning by someone who wanted the throne to go to another). When the city was taken, Marcellus supposedly gave orders to spare the scientist but Archimedes was killed nonetheless.

          Nine of Archimedes’s treatises, which demonstrate his discoveries in mathematics and in floating bodies, are extant. They are On the Sphere and Cylinder, On the Measurement of the Circle, On the Equilibrium of Planes, On Conoids and Spheroids, On Spirals, On the Quadrature of the Parabola, Arenarius (or Sand-Reckoner), On Floating Bodies, and On the Method of Mechanical Theorems. Archimedes’s many contributions to mathematics and mechanics include not only calculating the value of π but also devising a mathematical exponential system to express extremely large numbers (he said he could numerically represent the grains of sand that would be needed to fill the universe), developing the Archimedes principle, and inventing the Archimedes screw to move water from one elevation to another.

          Of particular importance to scientists and mathematicians in the 20^th and 21^st Centuries was the re-discovery of the Method of Mechanical Theorems manuscript, which had been thought to have been lost or stolen. In 1906 a palimpsest (a document written on recycled parchment, meaning the original material had been erased or otherwise removed) was brought to the attention of the great Danish philologist Johan Ludvig Heiberg, perhaps the greatest student of primary sources for classical Greek geometry, who studied it in Istanbul. He recognized the incompletely erased undertext of the parchment, which contained a collection of Greek Orthodox prayers, as the work of Archimedes, including portions of several works previously known from other manuscripts or translations, as well the Method, previously considered lost. Fortunately for science and history, Heiberg photographed the palimpsest because a few years later it disappeared in the fog of war surrounding World War I. But many of Archimedes’ words were illegible in the photos and many others were lost in the folds of the binding. Heiberg also had not copied Archimedes’s explanatory diagrams, which are crucial for understanding his thought processes.

          In October 1998, the manuscript surfaced again when Christie’s in New York City auctioned what they called the “Archimedes Palimpsest” to an unidentified American collector for $2,000,000. It was perhaps the most important single historical manuscript in the field of mathematics. The collector lent the manuscript to the Walters Art Museum in Baltimore, which is renowned for its rare book conservation department. Among many other things, analysis of the document revealed that Archimedes dealt with infinitely large sets, something that Greek mathematicians were reputed to avoid owing to its inherent difficulties.

In 2003-2004, Reviel Netz, a historian of science at Stanford University, and Nigel Wilson, a classics professor at Oxford University, concluded that Archimedes’s treatise, the Stomachion, was centuries ahead of its time. Netz and Wilson believe the work deals with combinatorics, a fiendishly difficult field involving combinations and permutations that did not come entirely into its own until the relatively recent rise of computer science.[2]

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[1] Source: Columbia Encyclopedia, 6^th Edition.

[2] For a fascinating article about Archimedes and the puzzle, see Erica Klarreich, “Glimpses of Genius Mathematicians and historians piece together a puzzle that Archimedes pondered,” Science News Online:

http://www.sciencenews.org/articles/20040515/bob9.asp and also see Stanford Report, November 6, 2002, found online at: http://news-service.stanford.edu/news/2002/november6/archimedes-116.html. See also Reviel Netz, Fabio Acerbi, and Nigel Wilson, (2004). Towards a Reconstruction of Archimedes’ Stomachion, Sciamus, 5, pp. 67-99.