Why did he change Newton's equations instead of Maxwell's?

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Why did Einstein decide that Maxwell's equations needed no modifying and that it was Newton's equations that were wrong? He noticed that Newton's equations allowed a speed of c, but that travelling at c will let you see EM waves that don't vary in time, hence violating Maxwell. So the options were that either Newton or Maxwell needed modifying. Why was it that Newton's equations needed changing so that c was constant in all reference frames?