ANNUITY ANOINTING 535 is the present value of the perpetuity, and the same discounted at compound interest for the intervals of the term w, at the assumed rate i (P fiTni ^*) ^ * ne P resen ^ value of the term annuity for n intervals. The subtrac- tive quantity (Q^ P, or vP as it is usually written) is called the reversionary value of the term annuity, or of the estate whose income is absorbed by its payment. Any annuity is said to be worth as many "years' purchase" as the times it is contained in its present value. The present value of any life annuity involves a medical as well as a mathematical question. The mathematical solution, which comes first, is founded on an assumed rate of mortality, more or less worthy of confidence, according to its agreement with observed facts when taken in large numbers. This gives the value supposing the life in question possessed of the average vitality due to its age. Medical science will modify this result so far as it can deter- mine a variation of the individual from the assumed average, though it has no means of fixing a definite or numerical variation. It is a common mistake to suppose that the present value of a life annuity can be found from the "expectation of life," or average after-lifetime at the given age, by finding the present value of an annuity certain for the term of that ex- pectation. This can be true only when the assumed interest is zero. This popular error has been much fostered by life insurance companies publishing tables of "expectation," which can have no possible application to their business except by this erroneous method, and which, so applied, only prove their premiums too high. The only correct method of applying the rates of mortality and interest to ascertain the present value of any series of payments contingent on life, is to apply them separately to each and every possible payment. Each future payment must, in effect, be multiplied by the present value at compound interest of a dollar, or monetary unit, payable certain at that time; and this product must again be multiplied by the fraction, derived from the table of mortality, expressing the probability of the party being then alive to pay it. The sum of as many such products as there are possible payments is the present value of the life annuity. When only one or two lives are concerned, there are tables which abridge the operation to a narrow compass; but when there are three or more, the combinations be- come too numerous to admit of exhaustive tables, and mathematicians content themselves with methods of approximation to solve par- ticular problems. After the mathematical solu- tion, which can only be as correct as the as- sumptions on which it is founded, comes the medical, weighing the special facts by which the individual case differs from the average or general type. The reason why the business of selling annuities to commence in a year or less is always unprofitable to an honest company, and why it is unprofitable to a government as a means of borrowing money, is that the med- ical selection is in favor of the buyer. On the contrary, companies dealing in policies of in- surance, or selling annuities long deferred, suc- ceed by a medical selection in their own favor. The most valuable recent contributions to the basis for calculating life annuities are con- tained in the works of Chisholm, and the writ- ings of Dr. Farr, in connection with the re- ports of the registrar general of Great Britain. Very valuable observations have also been made by Mr. Meech on the United States cen- sus of 1860, and by Mr. Elliott on the popula- tion returns of Massachusetts. ANNUNCIATION, the announcement to Mary by the angel (Luke i. 30-33) that she should conceive and bear the child Jesus. In com- memoration of this event, the church instituted the feast of the Annunciation, to be observed on the 25th of March. In old style this day commenced the year. Writers differ as to the time when this feast was instituted. Some throw it as far back as the 4th century, since there is mention of it in a sermon ascribed to Athanasius. Others think its origin is to be assigned to the 7th century, which is the most probable opinion, as the sermon of Athanasius is believed to be spurious. ANODYNE (Gr. av privative and bdbvri, pain), a term properly applied, not to medicines which relieve pain by removing its cause, but to those which merely diminish the conducting power of the nerves of sensation, or which render the brain less susceptible to or less conscious of pain. The principal medicines of this kind are opium, belladonna, and hyoscyamus, with their alkaloids, Indian hemp, ether, and chloroform. The last two are usually called anesthetics, because they diminish the power of the brain to receive impressions from any external source. The use of anodynes is generally to be looked upon as the substitution of a lesser evil for a greater, and a habitual resort to them is always, if possible, to be avoided. ANOINTING, an ancient custom of pouring aromatic oils on persons as a token of honor. It was employed in consecrating priests, proph- ets, kings, and the places and instruments appointed for worship. In the Old Testament, the anointed of the Lord is a person upon whom God has conferred a particular dignity, and whom he has appointed to a special minis- try. The anointing oil was often a very costly preparation. Olive oil, spikenard, and myrrh were the more common materials. A very precious oil, the holy oil, was used in the ser- vice of the sanctuary, and could not be applied to any ordinary purposes. The Roman Catholic church has retained anointing as a symbol in its sacraments of baptism, confirmation, ordina- tion, and extreme unction. In consecrating a church, the bishop anoints the walls of the edifice and the altars which are to serve in the celebration of the mass. Anointing with per- fumed oil was in common use among the
