What Lincoln is proposing here (as memorably acted by Daniel Day-Lewis, and brilliantly scripted by Tony Kushner) is that moral laws of justice and equality can be derived from the logic of geometry. What's so striking about this scene is that it so perfectly mimics Platonism and Pythagoreanism and their geometric foundations. Lincoln is making a worthwhile and commendable moral stand, but his reasoning is faulty. There's nothing in geometry that can actually answer, with any kind of logical finality, these moral questions. In spite of the worthiness of the cause, it amounts to an abuse of reason.
Joshua the way I would say that slightly differently is that Lincoln's argument shows how important it is to trace back one's reasoning to determine what the "first principles" are. Lincoln's first principle here is erroneous - he is explicitly citing Euclidian / Platonic universals. Even though it leads him in this case to a conclusion with which we agree. However because the first principle is wrong, his reasoning on other issues could go into other "absolutist" directions with which we would disagree.
I think this is why Epicurus was so concerned to establish confidence in an explicit and firm set of "first principles" (such as nothing comes or goes to nothing) and that he did in fact consider his principles of physics to be just such starting points for all other reasoning (specifically including ethics), as DeWitt is suggesting.
I remember the different perspectives that some of us had on this surfacing in the discussion of some of the recent "Reverence and Awe" issues. My view is that anyone who suggested that communicating with gods could be a part of Epicurean philosophy would forever be barred from successfully arguing that due to PD1, which serves as such an axiom or first principle. My view is that PD1 should be considered as forever ruling out such an approach in an Epicurean Philosophy context. Not everyone agreed that that line of reasoning would suffice as an absolute bar.
My thinking on that hasn't changed -- Epicurus intended (in my view) that the basic principles such as the 12 Principles of Nature and to a lesser extent (because they are more loose, such as at the end) be considered bedrock principles that serve that purpose of anchoring the philosophy in something firm, and that absent that anchor the philosophy would be just another person's set of assertions and entitled to no more deference. Identifying a set of bedrock principles anchors them in Nature rather than in Epicurus' personal preferences.