Monday, January 17, 2005

Using geometry to cheat on taxes

The last few chapters of yerushalmi sotah contain some interesting and amusing aggadic material. One such example occurs on yerushalmi Sotah 24a-b, perek 5 halacha 3:

The gemara is discussing the measure of migrash and techum of a city, where the migrash extends 1000 cubits from the city, and techum extends 2000 cubits. The definition of this extension, especially in terms of migrash, is non-trivial, but in the course of the discussion, use is made of the fact that if you compare two squares, one with a length (and therefore width) of A and the other with a length (and therefore width) of 2A, the area of the larger square will be 4 times rather than 2 times the area of the smaller square. This is because since you are maintaining the ratio of length : width so that it is a square, you double both length and width.

Image 1: Doubling the length (and width) of each side causing a quadrupling of the area of the resulting square
Thus, the area of the smaller square is A x A = A2 and that of the larger square is 2A x 2A = (2A)2= (2)2 x A2 = 4A2, that is, 4 times the area of the smaller square. In general, if you create a sqaure whose side is B times the side of a square with length A, the area will be B2 as large.

Which brings us to the gemara. After stating this principle in order to explain migrash and techum, it says
כהדא ריש גלותא
איטלק עילוי חד טריקלין ארבעין על ארבעין דימליניה חיטין
אתא לגבי רב הונא

אמר ליה פייסון דיסבון מינך עשרין על עשרין כדון.
ועשרין על עשרין בתר זמן ואת מיתגר פלגא
Like this story with the Exilarch
Who owed as tribute a room 40 by 40 {cubits} filled with wheat
He came before Rav Huna
He {Rav Huna} said to him: appease them that they will take from you 20 by 20 now
and 20 by 20 after a time, and you will thus profit half.

The room must have been a standard height. (If we were talking about a cube whose height also doubled, doubling a side would mean a volume of 2A x 2A x 2A = 23 x A3 = 8 x A3 which would mean that he would he would profit a lot more than half.)

We don't know whether the Resh Geluta took Rav Huna's advice, or what happened afterwards. But this seems to rely on the fact that the collectors, or else the Persian government, did not know their geometry.

This should not be taken as support for cheating on taxes.


Anonymous said...

hmmm, so what?

joshwaxman said...

good point.

really, though, I posted the picture in anticipation of the eventual post. hope to update today. Using geometry to cheat on taxes...

Anonymous said...

This is incorrect. You would have to double all sides to quadruple the area. Doubling two (opposite, or parallel) sides would double the area. Of course, this all assumes that the angles are maintained (and that we are dealing with a square in the Euclidean plane). If not, doubling a single side may result in quadrilateral whose area is less than double the original square.


joshwaxman said...

right. i meant doubling the measure of the side of square, which would mean doubling both length and width, would quadruple the area. I'll correct the terminology when I actually post.


joshwaxman said...

as should be clear from the picture

joshwaxman said...

note, to understand the earlier comments. Originally I just posted the image, with the caption: doubling the length of a side causing quadrupling of the area.

I was inexact in my language, and meant doubling the measure of each side in order to get a sqaure where each side was twice that of the original.


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