The present paper is a continuation of that in the Companion for 1840, on the application of Barrett's method of calculating Me contingencies. No tables were then published for contingencies involving two lives; but since that time Mr. Jones's tables have appeared, in his work on the subject in the Library of Useful Knowledge. It may therefore be worth while to collect and describe the formulæ relative to two lives which are most likely to be useful in practice; with some other matters, which may partly interest the actuary and partly the commencing; student.

The Calculus of Differences is a somewhat extensive subject, but it is not necessary now to go far into it, or to occupy ourselves with its application to other than the simplest class of functions, namely, rational, algebraical, and integer functions. The theorems we require might have been left to be deduced as occasion for them should arise; but it has been thought better— and the arrangement will certainly be more convenient for reference—to bring the greater portion of them together in this place.

In the Report before alluded to, this case is especially noticed as affording evidence of the concessions made by foreign countries to obtain uniformity. The practice abroad has hitherto conformed to the principle recognized universally, as to the propriety of admitting as general average the damage done by voluntary stranding which was attended by the preservation of the ship and cargo. They still retain, however, as a subject of contribution, any damage which may arise from running a ship ashore to avoid capture; but it is clear that the same objection cannot be raised as in the case of a vessel in danger of foundering; for she may be, under the former circumstances, perfectly seaworthy, and therefore there would not be the difficulty of determining the exact amount of damage done to the cargo or the ship.

Logic, though the other exact science, has not had the sort of assailants who have clustered about mathematics. There is a sect which disputes the utility of logic, but there are no special points, like the quadrature of the circle, which excite dispute among those who admit other things. The old story about Aristotle having one logic to trammel us, and Bacon another to set us free—always laughed at by those who really knew either Aristotle or Bacon—now begins to be understood by a large section of the educated world. The author of this tract connects the old logic with the indecencies of the classical writers, and the new with moral purity: he appeals to women, who, “when they see plainly the demoralizing tendency of syllogistic logic, they will, no doubt, exert their powerful influence against it, and support the Baconian method.” This is the only work against logic which I can introduce, but it is a rare one, I mean in contents, I quote the author's idea of a syllogism:—

“The basis of this system is the syllogism. This is a form of couching the substance of your argument or investigation into one short line or sentence—then corroborating or supporting it in another, and drawing your conclusion or proof in a third.”

This subject has been so fully treated by many writers, that it may be thought little or nothing new can be said in reference to it. It may, however, be useful to bring together and to compare the results obtained by the standard writers on the theory of life contingencies, to examine how far those results agree with each other, and when they differ, to ascertain the causes of the difference.