Gottfried Wilhelm Leibniz was born 50 years after Descartes (in 1646) and lived into the 18th century (until 1716). I’d recommend that you look at the introduction of the Garber and Ariew collection to get a good compact account of details of Leibniz’ life and his work, but let me note a few connections between him and Descartes.
Leibniz’ Discourse on Metaphysics, the work we will look at first, was his first mature philosophical work and was written when he was 40 (so at about the same age as Descartes was when he wrote the Discourse on Method). Also like Descartes, Leibniz’s early work was not only in philosophy but also in mathematics and physics. He seems to have developed the calculus at least somewhat independently of Newton (though after him), and his presentation of those ideas was sufficiently influential that it is Leibniz’s notation that became standard. And, while Descartes had argued for something like the conservation of momentum in physics, Leibniz was a key proponent of the principle of the conservation of kinetic energy (though the basic idea was due to the Dutch physicist Huyghens, 1629-1695).
During the 50 years between Descartes’ Discourse on Method and Leibniz’ Discourse on Metaphysics, Descartes’ work had become widely known and influential. Indeed, Leibniz was influenced not only by Descartes directly but also by way of other philosophers who were influenced by him. Notable among these were Antoine Arnauld (1612-1694), Baruch Spinoza (1632-1677), and Nicholas Malebranche (1638-1715). Arnauld and Malebranche could be described as Cartesians to some degree (though in different ways). Spinoza, who is the most important among the philosophers of the period that we won’t read, had a very individual position, but he developed it through a study of Descartes’s views. Malebranche would probably now be placed next after Spinoza in importance as an original philosopher of the time, but Arnauld was important mainly as a critic of others, initially Descartes (he wrote one of the series of objections published with the Meditations) and later of both Malebranche and Leibniz. And he was someone Leibniz respected: after writing his Discourse, he engaged in an extended correspondence with Arnauld in which he explained and defended his ideas.
Leibniz’ text is broken in many relatively short sections, so I won’t need to suggest ways of dividing it up, but I will suggest some larger groupings and the sort of topics they focus on.
• Sections 1-7 all concern in one way or another God’s ordering of the universe. What may not be clear here (though it will start to appear later in the work) is that the sort of order studied by physics is never far from Leibniz’ mind. The balance of simplicity and richness mentioned in §5 is the key idea in these sections, and it is in line with various maximum and minimum principles which had started to capture the imagination of physicists at this time (beginning with work of Fermat, 1601-1665, on refraction in 1662).
• The focus of the remaining sections in this assignment (§§8-13) is the idea of an individual substance. This is stated and developed in §§8-9, and the central problem it raises is addressed in §13. However odd it seems, don’t ignore it: it is probably the most central idea in Leibniz’ philosophy.