The Fifth Meditation opens with the Meditator turning his attention toward material objects. Rather than inquire into the things themselves, he inquires into her ideas regarding material things. He concludes that he can distinctly imagine extension, size, shape, position, and local motion, which is associated with duration.
The Meditator also considers that there are abstract geometrical objects which do not exist in the material world, do not depend on her mind, yet are not nothing. For instance, there are no triangles in the world, yet they have some kind of being. Even if no triangle has ever existed anywhere outside the mind of the Meditator, triangles still have a determinate essence which is independent of the Meditator's mind. The Meditator also denies that he has come to know the nature of triangles through the senses. After all, he can think up all sorts of shapes that he has never seen and derive their properties as clearly and distinctly as he does with those of the triangle. These properties must all be true since the Meditator clearly and distinctly perceives them. Besides, he notes, even before he began to doubt, he always regarded mathematical and geometrical objects as more certain than the objects of the senses.
The Meditator has reasoned that a triangle must have all the properties he ascribes to it, because the triangle exists as an idea in his mind and he clearly and distinctly perceives all these properties. He then reasons by analogy that God exists as an idea in his mind and he clearly and distinctly perceives all of his qualities. One of these qualities is existence, so it follows from his clear and distinct perception that God must exist. If existence is the essence of God, then God would not be God if he did not exist, just as a triangle would not be a triangle if it were not three-sided. At the very least, then, the existence of God must be as certain as the properties of mathematical and geometrical objects since he can prove them in the same way.
Clear and distinct perceptions are always convincing, according to the Meditator. Some perceptions may be evident, like the fact that a triangle has three sides, and some may take more thought, like the Pythagorean theorem that states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. But once the Pythagorean theorem is proved, it is just as certain as any other clear and distinct perception. Similarly with God: his existence would immediately be perceived clearly and distinctly if it weren't for the confusions caused by the senses and preconceived opinions. Now that God's existence has been established, it is as certain as any other clear and distinct perception.
The Meditator asserts that God is the guarantor of his clear and distinct perceptions. He admits that he cannot constantly fix his mental vision on any particular perception, so that there might be times when he is not clearly and distinctly perceiving a certain truth. At such times, doubt could creep in, if not for God. Because he knows that God is not deceiving him and has endowed him with a faultless understanding and a will than cannot but assent to clear and distinct perceptions, he knows that what he clearly and distinctly perceived in the past is and remains true even if he is not currently directing his mental vision toward it. Those judgments about which he is mistaken are not clearly and distinctly perceived by the intellect. And even if he is dreaming, as was suggested in the First Meditation, he cannot be mistaken with respect to a clear and distinct perception.
"Essence" is one of those philosophical terms thats currency has waned a bit since the seventeenth century. The essence of a thing is the property or set of properties that the thing cannot do without. For instance, Aristotle defines rationality as an essential characteristic of a human being. A person could lose a leg and still be human, but a person could not cease to be rational and remain human.