People often marginalize debunked scientific theories like Geocentrism and alchemy, but they are often rooted in deeply seated logic, contain many partial truths/good observations and remained in place because technology wasn't sufficient to provide much contrary evidence.
With this in mind, why would someone like Isaac Newton or any contemporary scientist believe in the existence of the philosopher’s stone? What “evidence” was there to convince people that transmutation or eternal life was not only possible, but readily attainable through alchemy?
You have to remember this isn't just some fringe view of the time (though it was starting to be looked down upon), alchemy was a popular subject at the time, and Newton and many others saw this as a science.
At this time, Chemistry was in its infancy as a Science, and in fact only started to take the shape of modern chemistry a few decades after his death, with the work of Antoine Lavoisier, that it came to resemble modern chemistry, as he defined the principle of mass conservation, and what an element was.
So alchemy was very much the precursor to this, as remember at this time, there was not the same understanding of the nature of atoms and molecules that later centuries would have, so they were trying to understand these things which they did not understand yet properly, and because of the misguided views that they had in the understanding of the nature and structures of elements, such as metals, it led to them believing they could make gold and performing experiments that they believed showed that minerals could grow.
Many even believed that metals were not elements, and instead compounds of sulfur and mercury or sometimes mercury, sulfur, and salt.
Can you see where they are coming from at this point. Without the proper understanding of such things, they were trying to find an understanding, with experiments, just like today. Many people see Alchemy as this pseudo-science they believed in back then, but in reality it is more like a misguided precursor which was required for modern chemistry to develop, and while it did not make gold, it did have some successes chemistry wise such as creating new alloys; manufacturing acids and pigments; inventing apparatus for distillation, the process used in making perfumes and whiskeys; conceiving of atoms centuries before modern atomic theory; and providing a template for the scientific method by running controlled experiments again and again.
Even Robert Boyle, (famous for Boyle's Law, which states that pressure is inversely proportional to volume in a closed system) around at the same time as Newton and seen as one of the first modern chemists, had much of his work grounded in alchemy. He encouraged doing this with a scientific method, where he assumed nothing in his experiments and compiled every piece of relevant data. As you can see it was treated as a proper scientific discipline, and it was only with the works of Lavoisier and John Dalton, with the rise of Modern Atomic Theory and the challenge to Phlogiston theory by Antoine Lavoisier (who proved that combustion requires a gas that has mass (oxygen) and could be measured by means of weighing closed vessels), that it was replaced by the modern chemistry we know now.
Also take into account that Newton was a religious person and a scientist, and he did write that the study of alchemy was a way to increase "the knowledge of god and secondly the way to find out true medicines"
http://discovermagazine.com/2010/jul-aug/05-isaac-newton-worlds-most-famous-alchemist#.Ux9vJvl_u9o