A followup on the "Pizza and Panini" piece. The "Pizza and Panini" piece is a sly comment on the Eurocentricity of the way we view contributions to science by the ancient Greeks versus the ancient Indians. Panini was, in my opinion, one of the greatest innovators in the ancient world. His grammar was the world's first formal system of language. Panini's ideas of formal rules in natural languages, in fact, significantly influenced the 19th and 20th century linguists who came after him - de Saussure's work (de Saussure,1894) and Chomsky's (Chomsky, 1957).

I am also poking a bit of fun at the lengthiness of some of the works of the ancient Greek mathematicians, scientists and even philosophers. Many of their dialogues appear unnecessary lengthy when viewed by us today. This is because the ancient Greeks had not yet developed the theories of languages, physics, et cetera that were developed after the European Enlightenment. If Euclid's propositions were analyzed today, we would find that they could have been written far more compactly. Two examples follow. The stuff in italics is all that would have been required for a Proof or Algorithm for the two Propositions of Euclid that I deal with below.

Anyway, pity about Panini. His name has not yet made it into curricula in America. I seldom use the term 'Eurocentricity' primarily because I think that it is not so much bias in America

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To cut off from the greater of two given unequal straight lines a straight line equal to the less.

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If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.

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1. Chomsky, Noam (1957), Syntactic Structures, The Hague/Paris: Mouton

2. Cours de linguistique générale, ed. C. Bally and A. Sechehaye, with the collaboration of A. Riedlinger, Lausanne and Paris: Payot; trans. W. Baskin, Course in General Linguistics, Glasgow: Fontana/Collins, 1977 (orig. c. 1894)

3. Ingerman, P. Z. (1967). ""Pāṇini Backus Form" suggested". Communications of the ACM 10 (3): 137.

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I am also poking a bit of fun at the lengthiness of some of the works of the ancient Greek mathematicians, scientists and even philosophers. Many of their dialogues appear unnecessary lengthy when viewed by us today. This is because the ancient Greeks had not yet developed the theories of languages, physics, et cetera that were developed after the European Enlightenment. If Euclid's propositions were analyzed today, we would find that they could have been written far more compactly. Two examples follow. The stuff in italics is all that would have been required for a Proof or Algorithm for the two Propositions of Euclid that I deal with below.

Anyway, pity about Panini. His name has not yet made it into curricula in America. I seldom use the term 'Eurocentricity' primarily because I think that it is not so much bias in America

*per se*that causes this sort of a thing. Instead, it is just a certain paucity of knowledge. People simply do not know a whole lot about ideas from outside America, especially about ideas from the distant past whose applicability for the here and now is limited. (Should they have to? Should they not have to? That is, of course, a very different set of questions). At the same time, of course, Americans are quite focused on the practical and a little bit of Eurocentricity never hurt anybody. In this case, there is a bit of Eurocentricity present because American syllabuses have not really been updated to reflect some of these sorts of contributions. Many, many college students in America have very likely never heard of him. And almost certainly, most would not be able to identify his very fundamental contributions. The term "Eurocentricity" is overused. So many things are attributed to it that the term has almost lost power and significance. But there is here a very strong case for Eurocentricity. So good luck to Panini. And I hope the term "Panini Backus form" gains far greater currency and that Panini gets to be somewhat more widely known than he is today. (Ingerman, 1967).=+=

__Book 1, Proposition 3__To cut off from the greater of two given unequal straight lines a straight line equal to the less.

**We are given 2 lines CZ & AB w/ |CZ| < |AB|.**__Given:____Algorithm:__*Describe circle w/ center A and radius |CZ| cutting AB at E.*__Result : Line required__*= BE*=+=

__Book 1, Proposition 4__If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.

**tr DEF and tr ABC w/ AB = DE & AC = DF. ang BAC = ang DEF.**__Given:____Proof.__*Superpose tr DEF on tr ABC with D superposed over A and E over B. Clearly, C will coincide with F since AC = DF and ang BAC = ang DEF. So, BC coincides with EF. The tr's coincide exactly. The result follows.*=+=

__REFERENCES__1. Chomsky, Noam (1957), Syntactic Structures, The Hague/Paris: Mouton

2. Cours de linguistique générale, ed. C. Bally and A. Sechehaye, with the collaboration of A. Riedlinger, Lausanne and Paris: Payot; trans. W. Baskin, Course in General Linguistics, Glasgow: Fontana/Collins, 1977 (orig. c. 1894)

3. Ingerman, P. Z. (1967). ""Pāṇini Backus Form" suggested". Communications of the ACM 10 (3): 137.

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__updated the text, added references.__**Update (June 2):**