What are Anselm's arguments for the existence of God?

Anselm developed an ontological argument, proving the existence of God through logic alone rather than evidence. He had two main arguments. The first rests on the predicate that God is 'a being than which nothing greater can be conceived' and continues as follows: everyone has an idea of God in their minds, even atheists, if then God exists even only as an idea it is possible that a greater being could exist outside of the mind, thus God must not just exist within the mind but also outside of it as well as He is the being than which nothing greater can be conceived. Anselm's second argument is that God either exists of He doesn't, if He exists then His existence must be necessary, if He does not then His existence must be logically impossible, God's existence is not logically impossible and therefore God must exist as a necessary being.Anselm's arguments are criticised directly by Gaunilo on two main fronts. The ontological argument is also taken up by Rene Descartes.