But it is fun trying to reconstruct an earlier free-wheeling approach. We will probably never know what the Greeks did in the missing period. Then the odd advanced book that expounds the Robinsohn concept of infinitesimal numbers. They would see a plethora of calculus books that all start from the limit concept. Imagine a far future civilization, that is researching our achievements in mathematics using only our current textbooks, missing the first 300 years. Archimedes was happy to go back to the ancient way operating, just to obtain informal results, but made it clear that the method was just suggestive. My thesis is that Greek mathematics went through a similar orbit to our calculus, and started with infinitesimals the history of this has been lost. However, we have almost nothing of Greek maths writing before Euclid, ~300 years of undocumented effort, Thales to Eudoxus. Specifically, Archimedes in “On Method”, thought of a sphere as an infinite collection of plane discs and weighed the discs using levers to arrive at. The conventional understanding is that Greek mathematics went the other way, starting with the method of exhaustion (a rigorous limit based method) and later using infinitesimals. You know how western calculus started with completed infinities, infinitesimals and infinite series, and how after many years and much heartache, these were replaced by the limit concept. Thinks: Here’s an opportunity to tangent again. The reader is invited to ponder other shapes and other dimensions. Plugging back in and setting gives the desired formula: So, calculating the volumes,ĭividing by and factoring the difference of two cubes, we then have Then the difference in the cones is approximately a cylinder of height, and with the same base as that of our original cone. That is, for all such cones we haveĪnd our problem is to find, the constant of proportionality. Whatever h, any such cone is similar, and so the volumes will scale as. Imagine a cone of height H, and imagine slicing the cone at some level, to give a new cone of height h. Here is Lodge’s argument, slightly reworded. Such arguments don’t vary all that much but, however we missed it, we’d never seen the derivation in the very elegant form presented by Lodge. Lodge’s note provides a simple derivation for the volume of a cone. Doing so, we stumbled across a “ mathematical note” from 1896 by Alfred Lodge, the first president of the Mathematical Association. A few days ago, we pulled on a historical thread and wound up browsing the early volumes of The Mathematical Gazette.
0 Comments
Leave a Reply. |