Here’s something I’d like to figure out, when, how and why it happened.<p>India has got “by” and “into” back to front: that is, they say “by” means <i>division</i> and “into” <i>multiplication</i>, whereas places like Australia and the USA have “by” meaning multiplication and “into” division.<p>I’ve confirmed this inversion with current students and with a retired chemistry teacher in his late 70s in Hyderabad, and with a man in his 30s in West Bengal.<p>If you ask WolframAlpha “3 by 4”, you get: “Assuming "by" is Times | Use Divide instead”. Ask it “3 into 4”, and you get: “Assuming "3 into 4" is referring to arithmetic | Use as a math function instead” and it does division (and the math function offered is `QuotientRemainder[4, 3]` which returns `{1, 1}`).<p>(Not sure if there’s an intrinsic reason to prefer either assignation. Full forms are commonly expressed “divided <i>by</i>” and “multiplied <i>by</i>”, and if “by” is just an abbreviation, <i>&c. &c.</i> Then there’s “of” which feels more definitely context-dependent: “paint three of the four albino elephants” is division, “I want three of those asbestos-plated helberds” is multiplication.)
For anyone interested, this 2-volume series is gold. The first volume deals with numerals and arithmetic, and the second with algebra, linear equations, quadratic forms, solutions to pell's equations, etc.<p><a href="https://archive.org/details/historyofhindumathematicsbhibutibhushandattaavadeshnarayansinghvol1and2_495_E/page/n7/mode/1up" rel="nofollow noreferrer">https://archive.org/details/historyofhindumathematicsbhibuti...</a>
My grandfather used to tell me what was written was known for centuries prior but handed down by word of mouth, all memorised by sages. It was only written down at a later date which became the point used of discovery for historians.