It may perhaps be thought by some persons, that the subject upon which I am to address you tonight, is one that requires no urging on your attention. It is admitted on all hands, I may be told, that a good actuary must necessarily go through a course of mathematical training, and nothing further need be said upon the point. But if an apology for the choice of my subject is required, I would remind you that a mathematician takes nothing for granted. Whereas many people will say that it is self-evident that “two and two make four”, and that this neither requires nor admits of proof, the mathematician knows that it is a proposition, which is not in the true sense of the word self-evident, but can be strictly proved. While we deal with simple matters of this kind, little harm is likely to result from assuming as self-evident, propositions which admit of proof. But the case is very different when we come to more complicated and obscure questions; and not the least advantage to be derived from a mathematical training, is the gradual forming of the habit of mind which requires strict proof to be furnished of many important propositions which are usually taken for granted. I purpose, then, instead of taking for granted the use of mathematical studies, to examine exactly why and how they are of service to the actuary; and I hope that the conclusions I shall arrive at, may be of some assistance to those among you who wish to extend their mathematical reading beyond the point that is by common consent considered indispensable to the actuary.