Monday, October 21, 2013

Location of the Yesod Ramp: A Mathematical Diversion


SUMMARY It is possible to determine mathematically where the minor ramp to the Yesod branched off from the Main Ramp based on the dimensions of the Altar.


Two minor ramps branched off of the Altar's Main Ramp, one on the west leading to the Yesod and the other on the east leading to the SovevTiferes Yisrael states that the "two minor ramps branched off near the foot of the Main Ramp, on its lower part" (Temple Diagram §48), which I take to mean that the minor ramps began somewhere on the lower half of the Main Ramp. In this post I attempt to figure out exactly where the ramp to the Yesod began.

The following diagram illustrates the relationship between the Altar, the Main Ramp, and the ramp to the Yesod. The Yesod ramp is represented by triangle XYZ and the Main Ramp by triangle ABC.

From the drawing we begin with:

This will ultimately take the form of a quadratic equation and can be rewritten as:


Since the ramp to the Yesod is assumed to conform to the Temple standard of 1 amah of height for every 3 amos of length, we can say that c = 3d such that:



From the diagram it is also true that triangle ABC is similar to triangle AYZ. Let side AB have the length l such that:


Substituting this result into the quadratic equation above yields:


The length of segment AD is known to be 30 amos, thus a = 30-e

 

This can be written out as:


The following known values may now be substituted:

f = 9 amos (height of the Main Ramp)
l = 32 amos (length of the Main Ramp)
b = 7 amos (distance from the corner of the Yesod to the Main Ramp)

which reduces the equation to the form:


Solving this quadratic equation for e will yield the distance from the foot of the Main Ramp that the ramp to the Yesod began. See what solution you come up with and check back next week to see how it compares to my answer and find out whether or not Tiferes Yisrael's contention that the Yesod ramp branched off on the lower half of the Main Ramp is supported by the math.

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