By D. S. Mitrinovic
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Extra info for Elementary Matrices - Tutorial Text No. 3
You can also easily measure the distance along the ground from where you’re looking to the bottom of the tree. From this information, trigonometry enables us to find the height of the tree. We know an angle, x, and the length of side adjacent to it, A; we’d like to find the length of the opposite side, O. The formulas tell us that Suppose we measure the angle to be 40°. 839. Suppose we also measured the distance A to be 10m (or, again, 10 yards or 10 of any other unit). 39m. As you can imagine, this was a very useful technique for ancient surveyors and builders to know about, and their successors still use it today.
In the book The Ascent of Man, the 20th-century mathematician and author Jacob Bronowski called it ‘the most important single theorem in all of mathematics’; that might be pushing it a bit, but it’s surely one of the ancient mathematicians’ great achievements. The first thing to notice is that on paper this looks like an equation about areas rather than lengths. After all, if A is a length, say 10cm (or 10in or 10 anything else), then A2 is the area of a 10cm × 10cm square, that is, 100cm2 (one To find the point X from the origin O, we walk a certain distance up and then to the right.
These methods were very useful but they seemed to rely on infinitesimal (infinitely small) distances in ways their inventors couldn’t really justify. What if the new physics of Newton and his followers was really based on an absurdity, like Zeno’s paradoxical argument? It was very worrying. So in between using calculus to solve mathematical problems and devising successful new physical theories, many people started asking how to make sense of those lurking infinities. The result was the idea of a limit.