Now, this itself does not prove that force equals mass times acceleration. But it suggests that this might be the case. You can form a hypothesis that it might be so, one you can test to show that indeed, force does equal mass times acceleration.
So, mathematics helps to identify relationships, which can help you formulate hypotheses to test and thus help develop theories in physics. Not just physics, but almost anything else.
This is how Newton worked and Principia masterfully shows how mathematics can be used to tease out the implications of your theories.
It also once more shows the vital role of calculus. Without which, Newton would not have been able to discover many of the Laws of Motion.
Newton observed that a prism refracts different colors of light at different angles. Which led him to conclude that color is a property intrinsic to light. Something which was then and is still sometimes hotly debated.
He then investigated the refraction of light and demonstrated that the multi-colored spectrum of light produced by shining light through a prism could be recomposed into white light by using a lens and another prism.
This showed that white light was in fact all the colors mixed together and that the prism merely served to separate them.
He was thus the first to understand the rainbow as the result of light being separated into different colors. Rain drops work rather like a prism. White light enters raindrops and the rain drops act much like a prism, separating out different colors and thus producing the range of colors seen in a rainbow.
He then proceeded to show that color is the result of objects interacting with colored light rather than objects generating the color themselves. Now we know that each color from the visible spectrum of light has a specific wavelength. And that the color of an object is determined by the wavelengths of light that gets reflected or absorbed when interacting with the object.
Newton’s work helped set the stage for this understanding and our understanding of how color and vision work.
His findings led him to conclude that colour is a property of the light, not a property of the objects themselves.
Before this it was customary to assume that color is an intrinsic property of objects. As if an apple is red because of some inherent property of redness in the apple. Or that the sky is blue because it has a inherent blueness property.