Several of my friends have suggested to me the propriety of publishing something of the kind now offered to the Society, observing, that the great brevity with which the Antecedental Calculus is written, and the very concise form in which it is delivered to the public, may lead some to form erroneous opinions respecting the principles on which it is founded. In compliance partly with their request, I have drawn up this short paper, which I hope will remove even the possibility of misconception on that head, and convince every intelligent reader, that the antecedental calculus has the same geometrical principles for its ground-work, that the formulæ in the Universal Comparison themselves have, from which I originally derived it more than twenty years ago.