LIBRATION

, of the Moon, is an apparent irregularity in her motion, by which she seems to librate, or waver, about her own axis, one while towards the east, and again another while towards the west. See Moon, and Evection. Hence it is that some parts near the moon's western edge at one time recede from the centre of the dise, while those on the other or eastern side approach nearer to it; and, on the contrary, at another time the western parts are seen to be nearer the centre, and the eastern parts farther from it: by which means it happens that some of those parts, which were before visible, set and hide themselves in the hinder or invisible side of the moon, and afterwards return and appear again on the nearer or visible side.

This Libration of the moon was first discovered by Hevelius, in the year 1654; and it is owing to her equable rotation round her own axis, once in a month, in conjunction with her unequal motion in the perimeter of her orbit round the earth. For if the moon moved in a circle, having its centre coinciding with the centre of the earth, whilst it turned on its axis in the precise time of its period round the earth, then the plane of the same lunar meridian would always pass through the earth, and the same face of the moon would be constantly and exactly turned towards us. But since the real motion of the moon is about a point considerably distant from the centre of the earth, that motion is very unequal, as seen from the earth, the plane of no one meridian constantly passing through the earth.

The Libration of the moon is of three kinds.

1st, Her libration in longitude, or a seeming to-andagain motion according to the order of the signs of the zodiac. This libration is nothing twice in each periodical month, viz, when the moon is in her apogeum, and when in her perigeum; for in both these cases the| plane of her meridian, which is turned towards us, is directed alike towards the earth.

2d, Her libration in latitude; which arises from hence, that her axis not being perpendicular to the plane of her orbit, but inclined to it, sometimes one of her poles and sometimes the other will nod, as it were, or dip a little towards the earth, and consequently she will appear to librate a little, and to shew sometimes more of her spots, and sometimes less of them, towards each pole. Which libration, depending on the position of the moon, in respect to the nodes of her orbit, and her axis being nearly perpendicular to the plane of the ecliptic, is very properly said to be in latitude. And this also is completed in the space of the moon's periodical month, or rather while the moon is returning again to the same position, in respect of her nodes.

3d, There is also a third kind of libration; by which it happens that although another part of the moon be not really turned to the earth, as in the former libration, yet another is illuminated by the sun. For since the moon's axis is nearly perpendicular to the plane of the ecliptic, when she is most southerly, in respect of the north pole of the ecliptic, some parts near to it will be illuminated by the sun; while, on the contrary, the south pole will be in darkness. In this case, therefore, if the sun be in the same line with the moon's southern limit, then, as she proceeds from conjunction with the sun towards her ascending node, she will appear to dip her northern polar parts a little into the dark hemisphere, and to raife her southern polar parts as much into the light one. And the contrary to this will happen two weeks after, while the new moon is descending from her northern limit; for then her northern polar parts will appear to emerge out of darkness, and the southern polar parts to dip into it. And this seeming libration, or rather these effects of the former libration in latitude, depending on the light of the sun, will be completed in the moon's synodical month. Greg. Astron. lib. 4, sect. 10.

Libration of the Earth, is a term applied by some astronomers to that motion, by which the earth is so retained in its orbit, as that its axis continues constantly parallel to the axis of the world.

This Copernicus calls the motion of libration, which may be thus illustrated: Suppose a globe, with its axis parallel to that of the earth, painted on the flag of a mast, moveable on its axis, and constantly driven by an east wind, while it sails round an island, it is evident that the painted globe will be so librated, as that its axis will be parallel to that of the world, in every situation of the ship.

LIFE-ANNUITIES, are such periodical payments as depend on the continuance of some particular life or lives. They may be distinguished into Annuities that commence-immediately, and such as commence at some future period, called reversionary life-annuities.

The value, or present worth, of an annuity for any proposed life or lives, it is evident, depends on two cir- cumstances, the interest of money, and the chance or expectation of the continuance of life. Upon the former only, it has been shewn, under the article ANNUITIES, depends the value or present worth of an annuity certain, or that is not subject to the continuance of a life, or other contingency; but the expectation of life being a thing not certain, but only possessing a certain chance, it is evident that the value of the certain annuity, as stated above, must be diminished in proportion as the expectancy is below certainty: thus, if the present value of an annuity certain be any sum, as suppose 100l. and the value or expectancy of the life be 1/2, then the value of the life-annuity will be only half of the former, or 50l; and if the value of the life be only 1/3, the value of the life-annuity will be but 1/3 of 100l, that is 33l. 6s. 8d; and so on.

The measure of the value or expectancy of life, depends on the proportion of the number of persons that die, out of a given number, in the time proposed; thus, if 50 persons die, out of 100, in any proposed time, then, half the number only remaining alive, any one person has an equal chance to live or die in that time, or the value of his life for that time is 1/2; but if 2/3 of the number die in the time proposed, or only 1/3 remain alive, then the value of any one's life is 1/3; and if 3/4 of the number die, or only 1/4 remain alive, then the value of any life is but 1/4; and so on. In these proportions then must the value of the annuity certain be diminished, to give the value of the like life annuity.

It is plain therefore that, in this business, it is necessary to know the value of life at all the different ages, from some table of observations on the mortality of mankind, which may shew the proportion of the persons living, out of a given number, at the end of any proposed time; or from some certain hypothesis, or assumed principle. Now various tables and hypotheses of this sort were given by the writers on this subject, as Dr. Halley, Mr. Demoivre, Mr. Thomas Simpson, Mr. Dodson, Mr. Kersseboom, Mr. Parcieux, Dr. Price, Mr. Morgan, Mr. Baron Maseres, and many others. But the same table of probabilities of life will not suit all places; for long experience has shewn that all places are not equally healthy, or that the proportion of the number of persons that die annually, is different for different places. Dr. Halley computed a table of the annual deaths as drawn from the bills of mortality of the city of Breslaw in Germany, Mr. Smart and Mr. Simpson from those of London, Dr. Price from those of Northampton, Mr. Kersseboom from those of the provinces of Holland and West-Friesland, and M. Parcieux from the lists of the French tontines, or long annuities, and all these are found to differ from one another. It may not therefore be improper to insert here a comparative view of the principal tables that have been given of this kind, as below, where the first column shews the age, and the other columns the number of persons living at that age, out of 1000 born, or of the age 0, in the first line of each column.|

These tables shew that the mortality and chance of life are very various in different places; and that therefore, to obtain a sufficient accuracy in this business, it is necessary to adapt a table of probabilities or chances of life, to every place for which annuities are to be calculated; or at least one set of tables for large towns, and another for country places, as well as for the supposition of different rates of interest.

Several of the foregoing tables, as they commenced with numbers different from one another, are here reduced to the same number at the beginning, viz, 1000 persons, by which means we are enabled by inspection, at any age, to compare the numbers together, and immediately perceive the relative degrees of vitality at the several places. The tables are also arranged according to the degree of vitality amongst them; the least, or that at Vienna, first; and the rest in their order, to the highest, which is the province of Vaud in Switzerland. The authorities upon which these tables de- pend, are as they here follow. The first, taken from Dr. Price's Observations on Reversionary payments, is formed from the bills at Vienna, for 8 years, as given by Mr. Susmilch, in his Gottliche Ordnung; the 2d, for Berlin, from the same, as formed from the bills there for 4 years, viz, from 1752 to 1755; the 3d, from Dr. Price, shewing the true probabilities of life in London, formed from the bills for ten years, viz, from 1759 to 1768; the 4th, for Norwich, formed by Dr. Price from the bills for 30 years, viz, from 1740 to 1769; the 5th, by the same, from the bills for Northampton; the 6th, as deduced by Dr. Halley, from the bills of mortality at Breslaw; the 7th shews the probabilities of life in a country parish in Brandenburg, formed from the bills for 50 years, from 1710 to 1759, as given by Mr. Susmilch; the 8th shews the probabilities of life in the parish of Holy-Cross, near Shrewsbury, formed from a register kept by the Rev. Mr. Garsuch, for 20 years, from 1750 to 1770; the 9th, for| Holland, was formed by M. Kersseboom, from the registers of certain annuities for lives granted by the government of Holland, which had been kept there for 125 years, in which the ages of the several annuitants dying during that period had been truly entered; the 10th, for France, were formed by M. Parcieux, from the lists of the French tontines, or long annuities, and verified by a comparison with the mortuary registers of several religious houses for both sexes; and the 11th, or last, for the district of Vaud in Switzerland, was formed by Dr. Price from the registers of 43 parishes, given by M. Muret, in the Bern Memoirs for the year 1766.

Now from such lists as the foregoing, various tables have been formed for the valuation of annuities on single and joint lives, at several rates of interest, in which the value is shewn by inspection. The following are those that are given by Mr. Simpson, in his Select Exercises, as deduced from the London bills of mortality.

The uses of these tables may be exemplified in the following problems.

Prob. 1. To find the Probability or Proportion of Chance, that a person of a Given Age continues in being a proposed number of years.—Thus, suppose the age be 40, and the number of years proposed 15; then, to calculate by the table of the probabilities for London, in tab. 1. against 40 years stands 214, and against 55 years, the age to which the person must arrive, stands 120, which shews that, of 214 persons who attain to the age of 40, only 120 of them reach the age of 55, and consequently 94 die between the ages of 40 and 55: It is evident therefore that the odds for attaining the proposed age of 55, are as 120 to 94, or as 9 to 7 nearly.

Prob. 2. To find the Value of an Annuity for a proposed Life.—This problem is resolved from tab. 2, by looking against the given age, and under the proposed rate of interest; then the corresponding quantity shews the number of years-purchase required. For example, if the given age be 36, the rate of interest 4 per cent, and the proposed annuity L250. Then in the table it appears that the value is 12.1 years purchase, or 12.1 times L250, that is L3025.

After the same manner the answer will be found in any other case falling within the limits of the table. But as there may sometimes be occasion to know the values of lives computed at higher rates of interest than those in the table, the two following practical rules are subjoined; by which the problem is resolved independent of tables.

Rule 1. When the given age is not less than 45 years, nor greater than 85, subtract it from 92; then multiply the remainder by the perpetuity, and divide the product by the said remainder added to 2 1/2 times the perpetuity; so shall the quotient be the number of years purchase required. Where note, that by the perpetuity is meant the number of years purchase of the fee-simple; found by dividing 100 by the rate per cent at which interest is reckoned.

Ex. Let the given age be 50 years, and the rate of interest 10 per cent. Then subtracting 50 from 92, there remains 42; which multiplied by 10 the perpetuity, gives 420; and this divided by 67, the remainder increased by 2 1/2 times 10 the perpetuity, quotes 6.3 nearly, for the number of years purchase. Therefore, supposing the annuity to be L100, its value in present money will be L630.

Rule 2. When the age is between 10 and 45 years; take 8 tenths of what it wants of 45, which divide by the rate per cent increased by 1.2; then if the quotient be added to the value of a life of 45 years, found by the preceding rule, there will be obtained the number of years purchase in this case. For example, let the proposed age be 20 years, and the rate of interest 5 per cent. Here taking 20 from 45, there remains 25; 8/10 of which is 20; which divided by 6.2, quotes 3.2; and this added to 9.8, the value of a life of 45, found by the former rule, gives 13 for the number of years purchase that a life of 20 ought to be valued at.

And the conclusions derived by these rules, Mr. Simpson adds, are so near the true values, computed from real observations, as seldom to differ from them by more than 1/10 or 2/10 of one year's purchase.

The observations here alluded to, are those which are founded on the London bills of mortality. And a similar method of solution, accommodated to the Breslaw observations, will be as follows, viz. “Multiply the difference between the given age and 85 years by the perpetuity, and divide the product by 8 tenths of the said difference increased by double the perpetuity, for the answer.” Which, from 8 to 80 years of age, will commonly come within less than 1/8 of a year's purchase of the truth.

Prob. 3. To find the Value of an Annuity for Two Joint Lives, that is, for as long as they both continue in being together.—In table 3, find the younger age, or that nearest to it, in column 1, and the higher age in column 2; then against this last is the number of years purchase in the proper column for the interest. Ex. Suppose the two ages be 20 and 35 years; then the value.

Prob. 4. To find the Value of the Annuity for the Longest of Two Lives, that is, for as long as either of them continues in being.—In table 4, find the age of the youngest life, or the nearest to it, in col. 1, and the age of the elder in col. 2; then against this last is the answer in the proper column of interest.—Ex. So, if the two ages be 15 and 40; then the value of the annuity upon the longest of two such lives,

N B. In the last two problems, if the younger age, or the rate of interest, be not exactly found in the tables, the nearest to them may be taken, and then by proportion the value for the true numbers will be nearly found.

Rules and tables for the values of three lives, &c, may also be seen in Simpson, and in Baron Maseres's Annuities, &c. All these calculations have been made from tables of the real mortuary registers, differing unequally at the several ages. But rules have also been given upon other principles, as by De Moivre, upon the supposition that the decrements of life are equal at all ages; an assumption not much differing from the truth, from 7 to 70 years of age.

Life-Annuities, payable half-yearly, &c.—These are worth more than such as are payable yearly, as computed by the foregoing rules and tables, on the two following accounts: First, that parts of the payments are received sooner; and 2dly, there is a chance of receiving some part or parts of a whole year's payment more than when the payments are only made annually. Mr. Simpson, in his Select Exercises, pa. 283, observes, that the value of these two advantages put together, will always amount to 1/4 of a year's purchase for half-yearly payments, and to 3/8 of a year's purchase for quarterly payments; and Mr. Maseres, at page 233 &c of his Annuities, by a very elaborate calculation, finds the former difference to be nearly 1/4 also. But Dr. Price, in an Essay in the Philos. Trans. vol. 66,| pa. 109, states the same differences only at 2/10 for half-yearly payments, and 3/10 for quarterly payments: And the Doctor then adds some algebraical theorems for such calculations.

Life-Annuities, secured by Land.—These differ from other life-annuities only in this, that the annuity is to be paid up to the very day of the death of the age in question, or of the person upon whose life the annuity is granted. To obtain the more exact value therefore of such an annuity, a small quantity must be added to the same as computed by the foregoing rules and observations, which is different according as the payments are yearly, half-yearly, or quarterly, &c; and are thus stated by Dr. Price in his Essay quoted above; viz, the addition is y/(2n) for annual payments, or b/(4n) for half-yearly payments, or q/(8n) for quarterly payments <*> where n is the complement of the given age, or what it wants of 86 years; and y, h, q are the respective values of an annuity certain for n years, payable yearly, half-yearly, or quarterly. And, by numeral examples, it is found that the first of these additional quantities is about 2/10, the second 1/10, and the 3d half a tenth of one year's purchase.

Complement os Life. See Complement.

Expectation of Life. See Expectation.

Insurance or Assurance on Lives. See Assurances on Lives.