"Groping" and "muddling through" is usually described as a solution by trial and error. ...a series of trials, each of which attempts to correct the … - George Pólya

<math>\frac {dy}{dx} = \frac {\omega^2x}{g}</math>...The first derivative, the result of the differentiation of <math>y</math> with respect to <math>x</math>, was written by Leibniz in the form
<math>\frac {dy}{dx}</math>...Leibniz's notation ...is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation... need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative <math>\frac {dy}{dx}</math> was considered as the ratio of two "infinitely small quanitites", of the infinitesimals <math>dy</math> and <math>dx</math>. ...it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure... it brought mathematics into disrepute... some of the best minds... such as... Berkeley, complained that calculus is incomprehensible. ...<math>\frac {dy}{dx}</math> is the limit of a ratio of <math>dy</math> to <math>dx</math>... Once we have realized this sufficiently clearly, we may, under certain circumstances, treat <math>\frac {dy}{dx}</math> so as if it were a ratio... and multiply by <math>dx</math> to achieve the separation of variables. We get
<math>{dy} = \frac {\omega^2x}{g}xdx</math>

Beauty in mathematics is seeing the truth without effort.

Here is a typical story about Mr. John Jones. Mr. Jones works in an office. He had hoped for a little raise but his hope, as hopes often are, was disappointed. The salaries of some of his colleagues were raised but not his. Mr. Jones could not take it calmly. He worried and worried and finally suspected that Director Brown was responsible for his failure in getting a raise. We cannot blame Mr. Jones for having conceived such a suspicion. There were indeed some signs pointing to Director Brown. The real mistake was that, after having conceived that suspicion, Mr. Jones became blind to all signs pointing in the opposite direction. He worried himself into firmly believing that Director Brown was his personal enemy and behaved so stupidly that he almost succeeded in making a real enemy of the director. The trouble with Mr. John Jones is that he behaves like most of us. He never changes his major opinions. He changes his minor opinions not infrequently and quite suddenly; but he never doubts any of his opinions, major or minor, as long as he has them. He never doubts them, or questions them, or examines them critically — he would especially hate critical examination, if he understood what that meant. Let us concede that Mr. John Jones is right to a certain extent. He is a busy man; he has his duties at the office and at home. He has little time for doubt or examination. At best, he could examine only a few of his convictions and why should he doubt one if he has no time to examine that doubt? Still, don’t do as Mr. John Jones does. Don’t let your suspicion, or guess, or conjecture, grow without examination till it becomes ineradicable. At any rate, in theoretical matters, the best of ideas is hurt by uncritical acceptance and thrives on critical examination. 2. A mathematical example. Of all quadrilaterals with