I need to write an essay in which I imagine the philosophers Descartes and Locke having dinner together. What would be some key topics for them to...

The most important step you need to take to outline this conversation is to establish what beliefs both men had in common, and how Descartes' modes of thinking about and perceiving the world might lead organically to Locke's political philosophy. In other words, because Descartes was the first major rationalist of the Enlightenment, who paved the way for all other Enlightenment philosophers with his brilliant use of thought experiments, you would first want to get Descartes and Locke to discuss their shared view of how to understand the world: through careful analysis and a the refusal to take accept any statement of fact on faith (also known as Skepticism).

Descartes and Locke could easily discuss Descartes contributions to mathematics, as well as his attempt to reconstitute humans'  understanding of the world through logic and methodical thought experimentation. Locke could very easily claim that Descartes' attempt to re-conceptualize the world led directly to Locke's attempt to re-conceptualize the assumptions about how governments worked, and to argue against absolute monarchies and for his new model of "The Social Compact/Contract."

These two men could also discuss the similarities between how the Social Compact reimagined political philosophy and how Cartesian geometry reimagined the physical world, by marrying geometry to algebra. Similarly, Locke's idea of the social compact brought together personal ethics, which had once been the sole province of religion, with theories of government.

You might even have Descartes suggest that Locke's social compact theory, which relied on citizens to consent to and adhere to laws of their own creation, was in reality a complex geometrical formula that requires reciprocity, which in geometric terms, might be compared to supplementary angles, which Descartes helped to measure, or the X and Y axis, neither of which works or can even exist in any sensible way without the other.

Remember that Descartes was a major contributor to our understanding of physics, and the social compact theory is a great example of a symbiotic relationship of two equal forces keeping one another in check. On one hand, citizens must agree to enact certain laws that govern their society, and on the other hand, those same citizens must submit to those laws that they have enacted, and be held accountable for any breach of those laws. This is a very rational, Cartesian idea, whose elegant construction Descartes would have loved.

You might also have the two philosophers compare Descartes' famous axiom which proved his existence, "Cogito Ergo Sum," (I think therefore I am,") with Locke's notion that, for a government to have legitimate power, its own people must both give it power and submit to that power. Both of these ideas rely on the notion of reciprocity and equality. In fact, the whole notion of equality, which is a balance of power and rights, is a mathematical equation.

Both Locke and Descartes understood the importance of natural equilibriums, because both were students and devotees of Aristotle, whom they would also certainly discuss. Without Aristotelian logic, neither Descartes nor Locke could have formulated their philosophies or world views. Both men and all of today's scientists and thinkers are descendants of Aristotelian analysis. So Locke and Descartes discussion should probably begin or end (or both) with a discussion of their chief intellectual influence: Aristotle.