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 of air upon the moistened surface. The manufacturers of common pins are obliged to keep their wire in a dry atmosphere, or immerged in water. If the wire be first moistened, and then exposed to the air, it will assume the brittle state much sooner. In this condition it breaks with a crystalline fracture, similar to that exhibited by an ingot. When a steel plate, such as a watch-spring, has been once tempered, the operation of simply rubbing it bright, will render it soft and elastic. The same change is brought about by slightly hammering it. It, however, resumes its elastic state, by being carefully heated till it becomes of a blue colour. If the heat be continued to redness, particularly in a close vessel, it becomes perfectly annealed.

have been treated of in the body of the work, but in such a manner as to render it necessary, that so useful a branch of knowledge should form the subject of an entirely new article in this place.

The doctrine of Compound Interest and Annuities-certain, is too simple ever to have occupied much of the attention of Mathematicians: inquiries into the values of interests dependent upon the continuance or the failure of human life, being more interesting and difficult, have occupied them more, but yet not so much as their importance would seem to demand; the discoveries, both in pure Mathematics and Physics, especially those of Newton, which distinguished the close of the seventeenth century, having provided them with ample employment of a more interesting kind, ever since the subjects of this article were submitted to calculation.

Fermat, Pascal, and Huygens, by laying the foundation of the doctrine of Probabilities about the middle of that century, first opened the way to the solution of problems of this kin. The earliest mathematical publication on Probabilities, the little tract of Huygens, De ratiociniis in ludo aleæ, appeared in 1658; and in 1671, his celebrated countryman, John De Witt, published a treatise on Life-annuities, in Dutch. (Montucla, Hist. des Math. Tom. III. p. 407.) This, however, appears to have been very little known, or read, and to have had no sensible influence on the subsequent progress of the science; the origin of which may be properly dated from the publication of Dr Halley’s paper on the subject, in the Philosophical Transactions for the year 1693 (No. 196.) That celebrated Mathematician there first gave a table of mortality, which he had constructed from observations made at Breslaw, and showed how the probabilities of life and death, and the values of annuities and assurances on lives, might be determined by such tables; which, he informs us, had, till then, been only done by an imaginary valuation. Besides his algebraical reasonings, he illustrated the subject by the properties of parallelograms, and parallelopipedons: there are, perhaps, no other mathematical inquiries, in the prosecution of which, algebra is entitled to so decided a preference to the elementary geometry as in these, and this example of the application of geometry has not been followed by any of the succeeding writers.

In the year 1724, Mr De Moivre published the first edition of his tract, entitled, Annuities on Lives. In order to shorten the calculation of the values of such annuities, he assumed the annual decrements of life to be equal; that is, that out of a given number of persons living at any age, an equal number dies every year until they are all extinct; and, upon that hypothesis, he gave a general theorem, by which the values of annuities on single lives might be easily determined: this approximation, when the utmost limit of life was supposed to be 86 years, agreed very well with the true values between 30 and 70 years of age, as deduced from Dr Halley’s table, and the method was of great use at the time; as no tables of the true values of annuities had then been calculated, except a very contracted one inserted by Dr Halley in the paper mentioned above. But, upon the whole, this hypothesis of De Moivre has probably contributed to retard the progress of the science, by turning the attention of Mathematicians from the investigation of the true law of mortality, and the best methods of constructing tables of the real values of annuities.

The same distinguished Analyst also endeavoured to approximate the values of joint lives; but it has since been found, that the formulæ he gave for that purpose are too incorrect for use. Mr Thomas Simpson published his Doctrine of Annuities and Reversions in the year 1742, in which the subject is treated in a manner much more general and perspicuous than it had been previously; his formulæ are adapted to any table of mortality, and, in the 7th corollary to his first problem, he gave the theorem demonstrated in the 149th number of this article, to which we owe ail the best tables of the values of life-annuities that have since been published.

In the same work, he also gave a table of mortality deduced from the London observations, and four others calculated from it, of the values of annuities on lives, each at three rates of interest; the first for single lives, the three others for two and three equal joint lives, and for the longest of two or of three lives.

These were the first tables of the values of joint lives that had been calculated; for although Dr Halley had shown, half a century before, how such tables might be computed, and had taken considerable pains to facilitate the work; the necessary calculations, by the known methods, previous to the publication of Mr Simpson’s Treatise, were so very 1em