{'fraction_non_alphanumeric': 0.024154589371980676, 'fraction_numerical': 0.0, 'mean_word_length': 4.524444444444445, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Symmetric Polynomials in the Work of Newton and Lagrange

A modern paraphrase of Newton’s remark might be that it is valuable to have “several different ways of looking at things.” Although we will work with quadratic equations instead of the cubics Newton asked about, the insight gained will be much the same, and the attendant computations will be less tedious. When we are finished you should be able to do the cubic for yourself, if you so desire, and compare your answer to Newton’s. In the course of our investigation, we will discover something that is worth much more than a simple solution to our problem. We will find evidence of this in another source, the work of Lagrange on the cubic and quartic equations. I have taught classes on the history of mathematics three times, and none of the students in any of those classes had encountered this material before. Indeed, I first encountered these ideas in a historical context, reading about the work of Newton. However, seventy or so years ago, when a course called the “Theory of Equations” was still common, this would probably not have been the case. Although the course is all but extinct, I hope that you will find these ideas from it to be natural, accessible, and exciting.