Archimedes

1. Archimedes Spiral
2. Archimedes Screw
3. Archimedes Painting
4. Archimedes Principle

Archimedes is a fossil that looks like a screw. It is a genus of fenestrate bryozoans, defined by a corkscrew-shaped axial support column and spiraling mesh-like fronds attached to the column. Broken fragments of Archimedes are common in Mississippian rocks of both eastern and western Kentucky. The fossils are named after the Greek scientist and philosopher Archimedes, who designed a corkscrew-shaped irrigation pump.

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Bryozoa are a phylum of marine invertebrate organisms that still live today. They are tiny (less than 1 millimeter long) organisms that construct colonial structures. Different types of prehistoric bryozoa built colonial mounds, branching bushes, crusts, and fan- or frond-like structures. Bryozoan structures look similar to the types of colonial structures built by corals, but they are a distinctly different phylum of organisms. Fenestrates are a type of bryozoan that had fan- and frond-shaped colonies. Fans and fronds were mesh-like—perforated by tiny holes.

He formulated the Archimedes' principle. One of the most famous contributions of Archimedes is.

Description. The genus name Archimedes is most commonly used to describe a corkscrew-shaped, bryozoan axial support column, although attached fronds are also defined under the genus. Columns are commonly fragmented. Most are a few millimeters wide and two to three centimeters in length, although longer columns are sometimes found. Archimedes structures appear to have been fragile and easily broken; therefore, whole three-dimensional fossils with complete fronds attached to axes are rare In addition to Archimedes frond genera, some species of the frond genera Fenestella and Fenestellina also are found attached to Archimedes support columns (although not all of the species of these genera were coiled and associated with Archimedes).

Species. At least 16 species of Archimedes, several with multiple varieties, have been reported from Kentucky (Butts, 1917; McFarlan, 1942; Conkin and Fuson, 1970; Chesnut and Ettensohn, 1988; Sable and Dever, 1990). Both the coiled axes and fronds may have species names (McFarlan, 1942; Bassler, 1953). Most of the species names assigned to Archimedes axial columns are based on variations in the shape of the convolutions (coils) and spacing of convolutions (McFarlan, 1942; Condra and Elias, 1944). Many species names were defined by Arthur C. McFarlan, a professor in the geology department (now the Department of Earth and Environmental Sciences) at the University of Kentucky, and past state geologist and director of the Kentucky Geological Survey. Some of the specimens he studied are in the Department of Earth and Environmental Sciences' paleontological collection. One species, found in the Mississippian Floyds Knob Bed of the Borden Formation, in Marion and Jefferson Counties, is called Archimedes kentuckiensis. The holotype, No. 164200, and many paratypes (Nos. 164201–164207) are in the U.S. National Museum collection (Conkin and Fuson, 1970).

Range and geographic occurrence. The oldest Archimedes fossils reported in Kentucky are from the Middle Mississippian Borden Formation, and they range through the uppermost Mississippian strata preserved in Kentucky. Archimedes fossils are very common in the Salem-Warsaw, Paint Creek, Golconda, Glen Dean, and Slade Formations, especially in shaly interbeds between limestones. Archimedes fossils are so common in the Warsaw Formation in the Illinois Basin (including western Kentucky) that historically the formation was called the 'Archimedes Limestone' (Willman, 1975).

Life and paleoecology. During the Mississippian Period, sea levels rose and fell many times, and Kentucky was sometimes covered by shallow tropical seas and at other times was a low-lying coastline. Archimedes thrived during the times Kentucky was covered by shallow tropical seas. Where Archimedes are found, their fossils are common to abundant, suggesting they lived in groups, sometimes called 'gardens' or 'thickets' on the sea floor. Archimedes commonly grew with crinoids and brachiopods in these thickets. Breakdown of fragile colonies also causes abundant fossil fragments where these fossils are found. Archimedes appears to have favored muddy seafloors, and perhaps the lee-side of shoals (McKinney, 1979; McKinney and Gault, 1980). Individual bryozoan zooids lined holes in the coiled fans, and filtered seawater through the fans for food (Cowen and Rider, 1972). Colonies were apparently capable of asexual reproduction, budding new fans from the fragments of broken fans (McKinney, 1983), which would have been an ecological advantage in soft bottoms and shallow, stormy seas in which thickets of Archimedes were periodically knocked over and fragmented.

References

• Bassler, R.S., 1953, Part G—Bryozoa, in Moore, R.C., ed., Treatise of invertebrate paleontology: Geological Society of America and University of Kansas Press, p. G120.
• Butts, C., 1918, Description and correlation of the Mississippian formations of western Kentucky: Kentucky Geological Survey, ser. 4, Geological Reports, 1918, 119 p.
• Chesnut, D.R., Jr., and Ettensohn, F.R., 1988, Hombergian (Chesterian) echinoderm paleontology and paleoecology, south-central Kentucky: Bulletins of American Paleontology, v. 95, p. 5–102.
• Condra, G.E., and Elias, M.K., 1944, Study and revision of Archimedes (Hall): Geological Society of America, Special Paper 53, 243 p.
• Conkin, J.E., and Fuson, M.L., 1970, Archimedes kentuckiensis, a new fenestrate bryozoan from the Floyds Knob Formation (Upper Osagean) of Kentucky: Journal of Paleontology, v. 44, p. 669–672.
• Cowen, R., and Rider, J., 1972, Functional analysis of fenestellid bryozoan colonies: Lethaia, v. 5, no. 2, p. 147–164.
• McFarlan, A.C., 1942, Chester bryozoa of Illinois and western Kentucky: Journal of Paleontology, v. 16, p. 437–458.
• McKinney, F.K., 1979, Some paleoenvironments of the coiled fenestrate bryozoan Archimedes, in Larwood, G.P., and Abbott, M.B., eds., Advances in bryozoology: London, Academic Press, p. 321–355.
• McKinney, F.K., 1983, Asexual colony multiplication by fragmentation: An important mode of genet longevity in the Carboniferous bryozoan Archimedes: Paleobiology, v. 9, no. 1, p. 35–43.
• McKinney, F.K., and Gault, H.W., 1980, Paleoenvironment of Late Mississippian fenestrate bryozoans, eastern United States: Lethaia, v. 13, p. 127–146.
• Sable, E.G., and Dever, G.R., Jr., 1990, Mississippian rocks in Kentucky: U.S. Geological Survey Professional Paper 1503, 125 p.
• Willman, H.B., 1975, Handbook of Illinois stratigraphy: Illinois State Geological Survey, Bulletin 95, 261 p.

At any rate, the method used by Archimedes differs from earlier approximations in a fundamental way. Earlier schemes for approximating pi simply gave an approximate value, usually based on comparing the area or perimeter of a certain polygon with that of a circle. Archimedes' method is new in that it is an iterative process, whereby one can get as accurate an approximation as desired by repeating the process, using the previous estimate of pi to obtain a new one. This is a new feature of Greek mathematics, although it has an ancient tradition among the Chinese in their methods for approximating square roots.

Archimedes' method, as he did it originally, skips over a lot of computational steps, and is not fully explained, so authors of history of math books have often presented slight variations on his method to make it easier to follow. Here we will try to stick to the original as much as possible, following essentially Heath's translation³.

The Approximation of Pi

The Key Theorem

If an angle of a triangle be bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and, if the segments of the base have the same ratio as the remaining sides of the triangle, the straight line joined from the vertex to the point of section will bisect the angle of the triangle.⁴

Archimedes' Method

The ratio of the circumference of any circle to its diameter is less than 3¹/₇ but greater than 3¹⁰/₇₁.

Proof:

I. Let AB be the diameter of any circle, O its center, AC the tangent at A; and let the angle AOC be one-third of a right angle.

Then
(1) OA : AC > 265 : 153

and

(2) OC : AC = 306 : 153.

First, draw OD bisecting the angle AOC and meeting AC in D.
Now

CO : OA = CD : DA

so that

(CO + OA) : CA = OA : AD

Therefore

(3) OA : AD > 571 : 153.

Hence

OD² : AD² > 349450 : 23409

Archimedes Spiral

so that

(4) OD : DA > 591¹/₈ : 153.

Secondly, let OE bisect the angle AOD, meeting AD in E.
Therefore

(5) OA : AE > 1162¹/₈ : 153

Thus

(6) OE : EA > 1172¹/₈ : 153.

Thirdly, let OF bisect the angle AOE and meet AE in F.
We thus obtain the result that

(7) OA : AF > 2334 ¹/₄ : 153

Thus

(8) OF : FA > 2339 ¹/₄ : 153.

He formulated the Archimedes' principle. One of the most famous contributions of Archimedes is.

Description. The genus name Archimedes is most commonly used to describe a corkscrew-shaped, bryozoan axial support column, although attached fronds are also defined under the genus. Columns are commonly fragmented. Most are a few millimeters wide and two to three centimeters in length, although longer columns are sometimes found. Archimedes structures appear to have been fragile and easily broken; therefore, whole three-dimensional fossils with complete fronds attached to axes are rare In addition to Archimedes frond genera, some species of the frond genera Fenestella and Fenestellina also are found attached to Archimedes support columns (although not all of the species of these genera were coiled and associated with Archimedes).

Species. At least 16 species of Archimedes, several with multiple varieties, have been reported from Kentucky (Butts, 1917; McFarlan, 1942; Conkin and Fuson, 1970; Chesnut and Ettensohn, 1988; Sable and Dever, 1990). Both the coiled axes and fronds may have species names (McFarlan, 1942; Bassler, 1953). Most of the species names assigned to Archimedes axial columns are based on variations in the shape of the convolutions (coils) and spacing of convolutions (McFarlan, 1942; Condra and Elias, 1944). Many species names were defined by Arthur C. McFarlan, a professor in the geology department (now the Department of Earth and Environmental Sciences) at the University of Kentucky, and past state geologist and director of the Kentucky Geological Survey. Some of the specimens he studied are in the Department of Earth and Environmental Sciences' paleontological collection. One species, found in the Mississippian Floyds Knob Bed of the Borden Formation, in Marion and Jefferson Counties, is called Archimedes kentuckiensis. The holotype, No. 164200, and many paratypes (Nos. 164201–164207) are in the U.S. National Museum collection (Conkin and Fuson, 1970).

Range and geographic occurrence. The oldest Archimedes fossils reported in Kentucky are from the Middle Mississippian Borden Formation, and they range through the uppermost Mississippian strata preserved in Kentucky. Archimedes fossils are very common in the Salem-Warsaw, Paint Creek, Golconda, Glen Dean, and Slade Formations, especially in shaly interbeds between limestones. Archimedes fossils are so common in the Warsaw Formation in the Illinois Basin (including western Kentucky) that historically the formation was called the 'Archimedes Limestone' (Willman, 1975).

Life and paleoecology. During the Mississippian Period, sea levels rose and fell many times, and Kentucky was sometimes covered by shallow tropical seas and at other times was a low-lying coastline. Archimedes thrived during the times Kentucky was covered by shallow tropical seas. Where Archimedes are found, their fossils are common to abundant, suggesting they lived in groups, sometimes called 'gardens' or 'thickets' on the sea floor. Archimedes commonly grew with crinoids and brachiopods in these thickets. Breakdown of fragile colonies also causes abundant fossil fragments where these fossils are found. Archimedes appears to have favored muddy seafloors, and perhaps the lee-side of shoals (McKinney, 1979; McKinney and Gault, 1980). Individual bryozoan zooids lined holes in the coiled fans, and filtered seawater through the fans for food (Cowen and Rider, 1972). Colonies were apparently capable of asexual reproduction, budding new fans from the fragments of broken fans (McKinney, 1983), which would have been an ecological advantage in soft bottoms and shallow, stormy seas in which thickets of Archimedes were periodically knocked over and fragmented.

References

• Bassler, R.S., 1953, Part G—Bryozoa, in Moore, R.C., ed., Treatise of invertebrate paleontology: Geological Society of America and University of Kansas Press, p. G120.
• Butts, C., 1918, Description and correlation of the Mississippian formations of western Kentucky: Kentucky Geological Survey, ser. 4, Geological Reports, 1918, 119 p.
• Chesnut, D.R., Jr., and Ettensohn, F.R., 1988, Hombergian (Chesterian) echinoderm paleontology and paleoecology, south-central Kentucky: Bulletins of American Paleontology, v. 95, p. 5–102.
• Condra, G.E., and Elias, M.K., 1944, Study and revision of Archimedes (Hall): Geological Society of America, Special Paper 53, 243 p.
• Conkin, J.E., and Fuson, M.L., 1970, Archimedes kentuckiensis, a new fenestrate bryozoan from the Floyds Knob Formation (Upper Osagean) of Kentucky: Journal of Paleontology, v. 44, p. 669–672.
• Cowen, R., and Rider, J., 1972, Functional analysis of fenestellid bryozoan colonies: Lethaia, v. 5, no. 2, p. 147–164.
• McFarlan, A.C., 1942, Chester bryozoa of Illinois and western Kentucky: Journal of Paleontology, v. 16, p. 437–458.
• McKinney, F.K., 1979, Some paleoenvironments of the coiled fenestrate bryozoan Archimedes, in Larwood, G.P., and Abbott, M.B., eds., Advances in bryozoology: London, Academic Press, p. 321–355.
• McKinney, F.K., 1983, Asexual colony multiplication by fragmentation: An important mode of genet longevity in the Carboniferous bryozoan Archimedes: Paleobiology, v. 9, no. 1, p. 35–43.
• McKinney, F.K., and Gault, H.W., 1980, Paleoenvironment of Late Mississippian fenestrate bryozoans, eastern United States: Lethaia, v. 13, p. 127–146.
• Sable, E.G., and Dever, G.R., Jr., 1990, Mississippian rocks in Kentucky: U.S. Geological Survey Professional Paper 1503, 125 p.
• Willman, H.B., 1975, Handbook of Illinois stratigraphy: Illinois State Geological Survey, Bulletin 95, 261 p.

At any rate, the method used by Archimedes differs from earlier approximations in a fundamental way. Earlier schemes for approximating pi simply gave an approximate value, usually based on comparing the area or perimeter of a certain polygon with that of a circle. Archimedes' method is new in that it is an iterative process, whereby one can get as accurate an approximation as desired by repeating the process, using the previous estimate of pi to obtain a new one. This is a new feature of Greek mathematics, although it has an ancient tradition among the Chinese in their methods for approximating square roots.

Archimedes' method, as he did it originally, skips over a lot of computational steps, and is not fully explained, so authors of history of math books have often presented slight variations on his method to make it easier to follow. Here we will try to stick to the original as much as possible, following essentially Heath's translation³.

The Approximation of Pi

The Key Theorem

If an angle of a triangle be bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and, if the segments of the base have the same ratio as the remaining sides of the triangle, the straight line joined from the vertex to the point of section will bisect the angle of the triangle.⁴

Archimedes' Method

The ratio of the circumference of any circle to its diameter is less than 3¹/₇ but greater than 3¹⁰/₇₁.

Proof:

I. Let AB be the diameter of any circle, O its center, AC the tangent at A; and let the angle AOC be one-third of a right angle.

Then
(1) OA : AC > 265 : 153

and

(2) OC : AC = 306 : 153.

First, draw OD bisecting the angle AOC and meeting AC in D.
Now

CO : OA = CD : DA

so that

(CO + OA) : CA = OA : AD

Therefore

(3) OA : AD > 571 : 153.

Hence

OD² : AD² > 349450 : 23409

Archimedes Spiral

so that

(4) OD : DA > 591¹/₈ : 153.

Secondly, let OE bisect the angle AOD, meeting AD in E.
Therefore

(5) OA : AE > 1162¹/₈ : 153

Thus

(6) OE : EA > 1172¹/₈ : 153.

Thirdly, let OF bisect the angle AOE and meet AE in F.
We thus obtain the result that

(7) OA : AF > 2334 ¹/₄ : 153

Thus

(8) OF : FA > 2339 ¹/₄ : 153.

Fourthly, let OG bisect the angle AOF, meeting AF in G.
We have then

OA : AG > 4673 ¹/₂ : 153.

Now the angle AOC, which is one-third of a right angle, has been bisected four times, and it follows that angle AOG = 1/48 (a right angle).
Make the angle AOH on the other side of OA equal to the angle AOG, and let GA produced meet OH in H.
Then angle GOH = 1/24 (a right angle).
Thus GH is one side of a regular polygon of 96 sides circumscribed to the given circle.
And, since

OA : AG > 4673 ¹/₂ : 153,

while

AB = 2 OA, GH = 2 AG,

it follows that

AB : (perimeter of a polygon of 96 sides) > 4673 ¹/₂ : 14688

But

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.

II. Next let AB be the diameter of a circle, and let AC, meeting the circle in C, make the angle CAB equal to one-third of a right angle. Join BC.

Then

AC : BC < 1351 : 780.

First, let AD bisect the angle BAC and meet BC in d and the circle in D. Join BD.
Then

angle BAD = angle dAC = angle dBD

and the angles at D, C are both right angles. It follows that the triangles ADB, BDd are similar.
Therefore

AD : BD = BD : Dd = AB : Bd
= (AB + AC) : (Bd + Cd)
= (AB + AC) : BC

or (BA + AC) : BC = AD : DB.

Therefore

(1) AD : DB < 2911 : 780.

Thus

(2) AB : BD < 3013 ³/₄ : 780.
Secondly, let AE bisect the angle BAD, meeting the circle in E; and let BE be joined. Then we prove, in the same way as before, that

(3) AE : EB < 5924 ³/₄ : 780 = 1823 : 240.

Therefore

(4) AB : BE < 1838 ⁹/₁₁ : 240.

Thirdly, let AF bisect the angle BAE, meeting the circle in F.
Thus,

(5) AF : FB < 3661 ⁹/₁₁ x ¹¹/₄₀ : 240 x ¹¹/₄₀
= 1007 : 66.

Therefore,

(6) AB : BF < 1009 ¹/₆ : 66.

Archimedes Screw

Fourthly, let the angle BAF be bisected by AG meeting the circle in G.
Then

AG : GB < 2016 ¹/₆ : 66, by (5) and (6).

Therefore

(7) AB : BG < 2017 ¹/₄ : 66.

Therefore BG is a side of a regular inscribed polygon of 96 sides.

It follows from (7) that

Archimedes Painting

(perimeter of polygon) : AB > 6336 : 2017 ¹/₄.

And .

Much more then is the circumference to the diameter

< 3 ¹/₇ but > 3 ¹⁰/₇₁.

Archimedes Principle