# Measuring Moon's elevation in units of length

The Sages taught in a baraita: If one witness says that he saw the moon two plow handles high above the horizon, and the other one says it was three plow handles high, their testimony is valid, as a small discrepancy of this kind is reasonable

How are they measuring Moon's elevation over the horizon in units of length (rather than angle)?

## mathematical background

It is impossible to measure elevation (altitude) of a planet over horizon in meters (feet, yards &c), just like it is impossible to measure time in kilograms.

To measure elevation (which is the angle between the horizontal direction and the direction to the celestial body in the vertical plane, one needs to measure two lengths: vertical and horizontal.

E.g., the observer has to say that the Moon's elevation was 2 meters high if the measuring stick is located 4 meters away from the observer, in which case the tangent of the elevation is 0.5(=2/4) and the elevation is ~0.46 radians or ~27 degrees. If the other witness claims that the elevation was 5 plow handles when the measuring stick was located 10 plow handles away, he has said the exact same thing (elevation ~27 degrees). If the 3rd witness reports that the elevation was 2 meters (same as 1st!) when the ruler was located 2 meters away (half of what the 1st used!), then he is saying that the elevation was 45 degrees which is significantly different from what the 1st observer says.

If, instead of the plow handles, the Baraisah mentioned fingers, it would have made sense because the assumption would have been that the distance to the measurement device (palm with fingers) is the length of the observer's arm, and the ratio of finger size to the arm's length is sufficiently close for most people to be within the general measuring error for a common person (although, let it be made clear that, 500 years before Mishnah, professional Babylonian astronomers made observations with precision of minutes).

• Unfortunately it does need further clarification. Im pretty sure trigonometry class was all about using "height" to calculate an angle. In this case the distance from moon to earth is a constant, so even if we don't know its value, two people standing in the same place at the same time ought to see the moon at the same height, and if they can't agree on the height, they're probably lying and never really saw the moon. Nov 3, 2021 at 3:52
• @Derdeer And how do you propose they should measure the distance they see in terms of plow handles in a consistent manner? Nov 3, 2021 at 4:01
• It is possible to do a measurement of the apparent height, but you have to have agreed on a consistent way to measure (eg place your measuring stick one meter away from you and measure the apparent height, and place your eye to the ground). Nov 3, 2021 at 4:11
• @sds They did not have the notion of angular displacement.
– pcoz
Nov 3, 2021 at 4:17
• @pcoz There are plenty of classical measurements of locations of celestial bodies in degrees, measured in various ways from various zeroes, already more than 2000 years ago. They had plenty of math available to them to more accurately describe what's going on here. But anyway, this is irrelevant, as one doesn't expect your average witness to be able to calculate such. Nov 3, 2021 at 12:47

Very much like the OP's answer, I suggest that the thickness of and distance to the plow handle is standard.

First of all, the thickness of a standard plow handle is stated in the Mishna (Kelim 17:8):

כָּל הַמִּטַּלְטְלִין מְבִיאִין אֶת הַטֻּמְאָה בָּעֳבִי הַמַּרְדֵּעַ, לֹא גָדוֹל וְלֹא קָטָן אֶלָּא בֵינוֹנִי. אֵיזֶה הוּא בֵינוֹנִי, כֹּל שֶׁהֶקֵּפוֹ טָפַח:

"Any movable object conveys uncleanness if it is of the thickness of an ox goad" it is one that is neither big nor small but of moderate size. What is meant by "one of moderate size?" One whose circumference is just a handbreadth.

(The commentators give different translations of מרדע, but that's beside the point.)

It seems like the reason for using this particular common object for measuring an observed height is because it is a long, flat, handheld item. One would hold it flat at arms length and estimate how many of them could be placed on top of each other between the horizon and the new moon.

The thickness of the handle then is 1/3 of a tefach (handbreath), approximately. The distance is a little less than two amos (cubits), let's say 10 tefachim. So the angles have tangents of about 2/30, 3/30 and 5/30, and the angles are 3.8, 5.7 and 9.5 degrees. Of course, none of these inputs are exact, but there is a very specific ballpark estimate.

One possible resolution (hinted at in the mathematical background of the question) is that the "plow handles" were more or less standard thickness and length. and

two plow handles high above the horizon

means 2 thicknesses over the length (which is reasonably low, as it should be for the new moon on the 1st night it appears -- which is why the alternative of 2 lengths over the length is unacceptable because this is far too high for the new moon on the 1st night it appears).

E.g., if a plow handle has length/thickness of ~10, then "two plow handles" means ~11.3 degrees and "5 plow handles" means ~26.6 degrees.