Important Articles

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Mathematics and Number Pi (3.14)

A Mechanical Derivation of the Area of the Sphere

by David Garber and Boaz Tsaban
Department of Mathematics and Computer Science
Bar-Ilan University, Israel, 2001
"In the beginning of the twelfth century CE, an interesting new geometry book appeared: The Book of Mensuration of the Earth and its Division, by Rabbi Abraham Bar Hiya (acronym RABH), a Jewish philosopher and scientist. This book is interesting both historically and mathematically. Its historical aspect is discussed in [3]. In this paper we consider the mathematical aspect. The second part of the book contains a beautiful mechanical derivation of the area of the disk [5, §95]. Roughly speaking, the argument goes as follows(see Figure 1): The disk is viewed as the collection of all the concentric circles it contains. If we cut the circles along the radius of the disk, and let them fan out to become straight lines, we get a triangle (because the ratio of the circumferences of the circles to their diameters is constant). The base length of the resulting triangle is equal to the circumference of the original circle, and its height is equal to the radius of this circle. Thus, the area of a circle is equal to half of the product of the radius and the circumference."
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The Weight of the Ark of The Covenant

by Elihu A. Schatz, PhD
Published in Jewish Bible Quarterly
Volume 35:2 April - June 2007
"The total weight obtained for the Ark, as stated above, is 83 kilograms (183 pounds), which consists of 5.0+14.7+11.9+25.0+20.0+2.6+0.03+4.2 kilograms. Each of the four men carrying the Ark would bear a weight of about 21 kilograms (46 pounds), which can easily be tolerated on the shoulders for extensive periods of time. If necessary, the carriers could use shoulder padding to ease the pressure of the poles on the shoulders."
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The Gold of the Ark

by Josia Derby, MA
Published in Jewish Bible Quarterly
Volume 33:4 October - December 2005
"To the weight of the gold of the Ark, we must add the weight of the lumber overlaid with gold, which we could estimate at 50 pounds, and the weight of the two Tablets of the Covenant at 60 pounds, for a grand total of 287.61 pounds. This would mean a weight of more than 70 pounds on a shoulder of each of four men. When a weight of this magnitude is distributed over an area, the pressure is inversely proportional to the area. Hence, the pressure from a pole two to three inches in diameter carrying such a weight would probably be unbearable even for a very strong man."
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The Mathematics of Levi ben Gershon, the Ralbag

by Shai Simonson
Department of Mathematics and Computer Science
Stonehill College
North Easton, MA 02067

"Levi comments: And if we say that the measurement around was taken on the inside of the basin, then that would be closer to the truth, but still just a close approximation. Since the width is an hand breadth , the diameter of the inner edge of the basin is ten cubits less one third of a cubit. And its circumference will be approximately one and one third cubits more than thirty cubits."
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Professor Isaac Elishakoff and Elliot M. Pines, PhD
Presented at the Fifth Miami International Conference on Torah and
Science, 16-18 December 2003
"Lost among the often abstract debates between Torah and science is the down-to-earth issue of π, the ratio between the circumference and diameter of a circle. Tanakh (Bible) and mathematics appear at odds over this simple constant. Although we all learned in school that π equals 3.141592…, it appears that the Tanakh claims 3 as an exact or at least approximate value of π."
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Robert K. Moniot, Associate Professor
Faculty of Arts and Sciences
Fordham University  
"In this section of the book Vitruvius is describing the best way to space the columns around a temple. The passage quoted here is addressing how to construct a temple whose length is twice its breadth. If the columns are equally spaced, the solution is to subtract one from the number of columns along the breadth (giving the number of intercolumniations), double that, and add one to yield the number of columns along the length."
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Engineering and Architecture

English translation by Bill Thayer
"In the aræostylos it is only necessary to preserve, in a peripteral building, twice the number of intercolumniations on the flanks that there are in front, so that the length may be twice the breadth. Those who use twice the number of columns for the length, appear to err, because they thus make one intercolumniation more than should be used."
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Americans with Disabilities Act (ADA)
U.S. Architectural and Transportation Barriers
Compliance Board (Access Board)
1331 F Street, N.W., Suite 1000
Washington, D.C. 20004-1111
"4.8 Ramps.
4.8.1* General. Any part of an accessible route with a slope greater than 1:20 shall be considered a ramp and shall comply with 4.8.
4.8.2* Slope and Rise. The least possible slope shall be used for any ramp. The maximum slope of a ramp in new construction shall be 1:12."

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Chemistry and Material Science

Characteristics of briarwood

by G. Tsoumis, N. Kezos, I. Fanariotou, E. Voulgaridis and C. Passialis
Department of Forestry and Natural Environment
Aristotelian University
Thessaloniki 54006, Greece
The density and hardness of briarwood are a little higher than that of stem wood, volumetric shrinkage is higher, but directional shrinkage tends to be isometric. Extractive content is very high. Ash is higher, and silica low. Boiling in water (removal of extractives) reduced the volumetric shrinkage of briarwood (the effect is normally opposite), did not affect the drying rate, but resulted in higher dimensional stability, and eliminated drying defects. Exposure to high temperatures (150-600°C) showed a higher resistance of briarwood, which could be attributed to extractives.
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Human Anthropometric Data and Physical Fitness Information

Fire Apparatus Manufacturer’s Association
FAMA Technical Committee
Chassis Subcommittee
Roger Lackore - Pierce Manufacturing
October 20, 2007

"The final cleaned database consists of 737 subject measurements of which 682 are male and 55 are female. The average subject age is 38, with a standard deviation of 9, a maximum of age of 68, and a minimum of age of 16. Results are given for the combined population as well as male and female populations independently." 
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Livestock and Agriculture

Building an Ox Yoke

by David Kramer
Tillers International, 1997
10515 E. OP Ave.
Scotts, Michigan 49088 USA

"A neck yoke rests directly on the necks of the oxen. It is carved and rounded to fit the animals' necks comfortably. Bows pass under the necks of the oxen and secure them in position. The oxen transfer their power to the yoke beam by pushing with the top of their neck directly against the beam, and by pushing with their shoulders against the upper part of the bows."
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Dairy Cow Comfort -  Free-stall Dimensions

by Neil Anderson,
Veterinary Science, OMAF,
Fergus, Ontario
January 2008

"Table 1 shows measurements of mature Canadian Holsteins and some calculated proportions, taken at a local herd. For example, the cows had a rump height of 60 inches, a nose-to-tail length of 8.5 feet, and a hook bone width of 25 inches. Their weight exceeded 1550 pounds."
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Oxen Logging

by Mark Seymour
FTP International Ltd.
P.O. Box 484
00101 Helsinki, Finland

"The working life of an ox is about 8 years, between the ages of 2.5 to 10.5. The length of a working day for logging oxen is normaly 5-6 hours."

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Modernization of Draught Animal Power

by N.S Ramaswamy, Ph.D.
Indian Heritage Academy
20th Main, 6th Block,
Koramangala, Bangalore, 56 00 95

"Double Animal Cart – 1.5 Tonne and 2 Tonne capacity. These are used in villages and towns for carrying medium loads and where there are not many ups and downs and the track is more or less even."

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Estimating Ox-Drawn Implement Draft

by Tim Harrigan, Richard Roosenberg, Dulcy Perkins and John Sarge
The authors are Ast. Professor, Agricultural Engineering Dept., Michigan State University, East Lansing, MI and Executive Director, Shop Coordinator and Project Coordinator, respectively, Tillers International, Kalamazoo MI.
"Well-conditioned oxen are capable of working draft loads equal to 10-12% of their body weight throughout the day and greater loads for short periods of time. This bulletin provides a range of normal draft requirements for tillage implements, wagons, sleds, stoneboats and some commonly used logging implements. This information can help the teamster plan fieldwork, match implements with the power available and establish realistic guidelines for animal performance."

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Religion and Tradition

Electronic Text Center, University of Virginia Library, 1995
*(this deuterocanonical book is also known as "The Book Of Wisdom", and it is mentioned by Nahmanides in his commentary)

"9:8: Thou hast commanded me to build a temple upon thy holy mount, and an altar in the city wherein thou dwellest, a resemblance of the holy tabernacle, which thou hast prepared from the beginning."

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by Rabbi Dov Kramer
Toras Aish Terumah 5768
Volume XV Number 22
Page 6

"The Lekach Tov also says (26:9) that the extra (sixth) section consisted of two cubits that doubled over in the front. This is also consistent with the BdMhM, while specifying that the extra 4 cubits in front did not hang down, but were folded over into a double-layer, and were "like a veil (or similar ornament) on the forehead" (keep in mind that there were other coverings above this second layer). Even though the BdMhM did not have the lower covering laying on top of the posts by the entrance, it would seem that this double-thick second layer did. This extra section did not hang down at all; rather, after extending two cubits beyond the lower covering was folded over, with the second two cubits folded back, towards the back of the Mishkan.
True, it wasn't literally "over the back of the Mishkan," but as opposed to hanging down, this "half" of the "extra section" "extended towards the back of the Mishkan." As the Lekach Tov puts it (26:12), "for half was folded towards the front of the Tent, and the two cubits - the [cubits] remaining - extended over (read: towards) the back of the Mishkan."
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by Rabbi Dov Kramer
Toras Aish Vayakhel 5768
Volume XV Number 25
Page 6

"As a matter of fact, Rav Chaim Kanievsky, shlita, in his comments on the Beraisa, says that the Beraisa understands the verse placing the "paroches"under the clasps as referring to the brass clasps of the second layer, not the gold clasps of the lower layer."

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by Rabbi Dov Kramer
Toras Aish Pekudei 5768
Volume XV Number 26
Page 1

"The Chizkuni (26:5) says that the 50 loops through which clasps went took up 3.5 of the 28 cubits of the width of the lowest covering, making the part of the clasp that fit into each loop either slightly more than a third of a tefach (if there are 5 tefachim/cubit) or slightly more than 4 tenths of a tefach (if there are 6 tefachim/cubit). With the Malbim's assertion (26:4-5) that the clasps were "on" one of the edges of the covering while being "at the edge" of the other, if the clasps started from the eastern edge, we can count both ends of the clasp (the part that went through the loops of the eastern part and the part that went through the loops of the western part) as well as the stem of the clasp that connected it's two fasteners, putting the edge of the Paroches almost another tefach closer to the edge of the clasps."
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by Rabbi Dov Kramer
Toras Aish Pekudei 5765
Volume XII Number 22
Page 1

"To illustrate (pardon the pun) the difficulty in creating a fully accurate representation of the Mishkan, let's look at just one part of it—the curtains that surrounded the courtyard.
For starters, there is a dispute (Zevachim 59b) about how tall the curtains were, 5 amos (cubits) or 15 amos, and whether the screen that covered the entranceway was 5 amos or 20 amos tall. But there are other issues that affect how the courtyard's enclosure should be drawn."
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by Rabbi Dov Kramer
Toras Aish Vayakhel 5765
Volume XII Number 25
Page 3

"Although there are many more than three approaches, as far as addressing the specific issue of having one less space than necessary, they all fall under one of these "categories" of approaches - with the variances having a minimal effect (if at all) on how to get to 100 cubits (and 50 on the east and west sides). Therefore, for simplicity sake, I will refer only to these three approaches, and explain why it would be difficult to claim that Rashi meant any one of them."
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by Rav Meir Spiegelman
Translated by Kaeren Fish
The Israel Koschitzky Virtual Beit Midrash
Yeshivat Har Etzion
Alon Shvut, Israel, 90433

"First of all, the boards involve a weight problem. If the board is a cubit thick from top to bottom, and it is made of solid wood, then the volume of each board is much more than a cubic meter of wood. Based on a conservative estimate that the specific weight of wood is half of the weight of water (in fact, different types of wood have different weights), we conclude that each board weighs more than half a metric ton. Since the Mishkan included at least fifty boards, we must arrive at the fantastic total of around twenty-five tons (not including poles, the boards of the screen, the parokhet, etc.). The Torah states that two wagons served to transport the boards. Thus, if our calculations are correct, each wagon – drawn by only two oxen – must have borne more than twelve tons. Clearly, this is not physically possible, but nowhere is there any indication that the transportation of the boards in the wagons was in any way miraculous. If, on the other hand, we assume that the boards stood in the shape of a split pyramid, their weight is halved, which seems more reasonable."

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A Dictionary of the Bible. Ed. J. Hastings.
Vol. 4. New York: Charles Scribner's Sons
"To provide the necessary rigidity for the frames the simple device is adopted of running five wooden bars along the three sides, passing through rings attached to the woodwork of the frames. Much needless discussion has been raised over the expression 'the middle bar in the midst of the boards (v. 28), which has been taken by various writers to mean that the middle bar of the five is intended to pass from end to end through a hole pierced in the heart of the massive 'boards' of the traditional theory (see diagrams of Riggenbach, Brown, etc. ). But the phrase is merely an epithet, after P's well-known manner, explanatory of the bar in question, the distinguishing feature of which is that it runs along the whole length of its side, north, west, south, as the case may be, in contradistinction to the remaining four, which we may presume run only half-way along-one pair at the top, the other pair at the bottom of the frames. This arrangement of the bars suggests that the frames were provided with three cross-rails--one at the top, rounded like the ends of the uprights to avoid injury to the curtains, another in the middle, and a third immediately above the bases. We thus obtain a double row of panels right round the dwelling (see the accompanying illustration with drawings to scale from a specially prepared model)."
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by Seth Mandel
Toras Aish Avodah
Volume 12, Number 65
December 26, 2003
"In summary, all depictions of the Menorah in the Hekhal, from Jewish and Roman sources from the time of the BhM and the following couple of centuries (as well as medieval Jewish sources) show the Menorah with rounded arms. (It is worth mentioning that the 4 Medieval Jewish drawings of the Menorah in MY pp. 69-72 all show arms like in a) and b): curved at the bottom, but straight in the upper part.)"
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