Patrick Henry was one of the leading orators of the Revolutionary generation, and in this speech, he certainly makes use of both appeals to reason and appeals to emotion. (Indeed, his argumentation here, even when it is logical, often has a highly emotional component to it.) He sees no room for compromise on this subject. This he establishes early on, when he declares it to be "a question of freedom or slavery." This, in itself, is a powerful piece of rhetoric: Henry frames this question around a clear bifurcation, organized around moral lines and extraordinary stakes.
Henry supports his position by referencing recent history with Great Britain:
Ask yourselves how this gracious reception of our petition comports with those warlike preparations which cover our waters and darken our land. Are fleets and armies necessary to a work of love and reconciliation? Have we shown ourselves so unwilling to be reconciled that force must be called in to win back our love? Let us not deceive ourselves, sir. These are the implements of war and subjugation; the last arguments to which kings resort.
This passage is simultaneously an appeal to reason and to emotion. It appeals to reason by way of its evidentiary nature. At the same time, however, his conclusion is blatantly incendiary in its effect. If it is established that Britain is preparing war against its colonies, this claim carries with it a highly emotional component, from which might arise feelings such as anxiety, outrage, fear, etc. Thus, this passage combines both pathos and logos to powerful effect.
He proceeds to further advance this argument, asking whether there can be any alternative explanation for Britain's recent actions. He answers in the negative.
For Patrick Henry, then, peaceful negotiations have already proven to be a failed solution. Moreover, however, in painting Britain as a tyrannical actor, he can frame armed rebellion as a moral choice.