Thus, in Dr HALLEY'S TABLE, in the article ANNUITIES, Sec. II. the fum of all the living at 20 and upwards is 20.724, which, divided by 598, the number living at the age of 20, and half unity fubtracted from the quotient, gives 34.15 for the expectation of 20. In calculating the value or expectation of joint lives, M. De Moivre fell into various miftakes, which we need not enumerate; as Dr Price and Mr Morgan have given tables of the value of lives, not founded on any hypothefis, but deduced from bills of mortality. Mr Morgan has likewife given rules for calculating values of lives in this manner. Dr Price, in the 3d effay in the first volume of his Treatise on Reversionary Payments, has alfo given proper rules for calcula ting thefe values, the most important of which are comprehended in the following paragraphs.

(2)SURVIVORSHIP, CASES OF. I. Suppofe a fet of married men to enter into a fociety in order to provide annuities for their widows, and that it is limited to a certain number of members, and couftantly kept up to that number by the admiffion of new members as the old ones are loft; it is of importance, in the first place, to know the number of annuitants that after fome time will come upon the establishment. Now, fince every marriage produces either a widow or widower, and fince all marriages taken together would produce as many widows as widowers, were every man and his wife of the fame age, and the chance equal which shall die firft; it is evident, that the number of widows that have ever exifted in the world, would in this cafe be equal to half the number of marriages. And what would take place in the world muft alfo, on the fame fuppoAtions, take place in this fociety. In other words, every other person in such a society leaving a widow, there muft arife from it a number of widows equal to half its own number. But this does not determine what number, all living at one and the fame time, the fociety may expect will come to be conftantly upon it. It is, therefore, neceffary to determine how long the duration of furvivorship between perfons of equal ages will be compared with the duration of marriage. And the truth is, that, fuppofing the probabilities of life to decrease uniformly, the former is equal to the latter; and confequently that the number of furvivors (or, which is the fame, fuppofing no fecond marriages), of widows and widowers alive together which will arife from any given fet of fuch marriages conftantly kept up, will be equal to the whole number of marriages; or half of them (the number of widows in particular) equal to half the number of marriages. Now it appears that in moft towns the decrease in the probabilities of life is in fact nearly uniform. According to the Breslaw Table of Obfervation, (fee ANNUITIES, Sec. II.) almoft the fame numbers die every year from 20 years of age to 77. After this Indeed, fewer die, and the rate of decrease in the probabilities of life is retarded. But this deviation from the hypothefis is inconfiderable; and its effect, in the prefent cafe, is to render the duration of furvivorship longer than it would other wife be. According to the London Table of Obfervations, the numbers dying every year begin to grow lefs at 50 years of age; and from hence to

extreme old age there is a conftant retardation in the decreafe of the probabilities of life. Upon the whole, therefore, it appears that, according to the Breflaw Table, and fuppofing no widows to marry, the number inquired after is fc newhat greater than half the number of the fociety; but according to the London Table, a good deal greater. This, however, has been determined on the fuppofition that the husbands and wives are of equal ages, and that then there is an equal chance who fhall die firft. But in reality husbands are generally older than wives, and males have been found to die fooner than females, as appears incontestably from several of the tables in Dr Price's Treatise on Reversions. It is therefore more than an equal chance that the husband will die before his wife. This will increase confiderably the duration of furvivorship on the part of the women, and confequently the number which we have been inquiring after. The marriage of widows will diminish, this number, but not fo much as the other caufes will increase it. II. If the fociety comprehends in it from the firft all the married people of all ages in any town, or among any class of people where the numbers always continue the fame, the whole collective body of members will be at their greatest age at the time of the eftablithment of the fociety; and the number of widows left every year will at a medium be always the fame. The number of widows will increase con. tinually on the fociety, till as many die off every year as are added. This will not be till the whole collective body of widows are at their greatest age, or till there are among them the greateft pof. fible number of the oldeft widows; and therefore not till there has been time for an acceffion to the oldest widows from the youngest part. Let us, for the fake of greater precifion, divide the whole medium of widows that come on every year into different claffes according to their different ages, and fuppofe some to be left at 56 years of age, fome at 46, fome at 36, and fome at 26. The widows, conftantly in life together, derived from the first clafs, will come to their greatest age, and to a maximum, in 30 years, fuppofing, with M. de Moivre, 86 to be the utmost extent of life. The fame will happen to the second class in 40 years, and to the third in so years. But the whole body compofed of thefe claffes will not come to a maximum till the fame happens to the fourth or youngest clafs; that is, not till the end of 60 years. After this the affairs of the fociety will become ftationary, and the number of annuitants upon it of all ages will keep always nearly the fame. III. If a fociety begins with its complete number of members, but at the fame time admits none above a particular age; if, for inftance, it begins with 200 members all under 50, and afterwards limits itself to this number, and keeps it up by admitting every year, at all ages between 26 and 50, new members as old ones drop off; in this cafe, the period neceffary to bring the maximum of annuitants will be just doubled. To determine the fum that every individual ought to pay in a fingle prefent payment, in order to entitle his widow to a certain annuity for her life, let us fuppofe the annuity 31. per annum, and the rate of interest.four per cent. It is evident, that the

value of fuch an expectation is different, accord. ing to the different ages of the purchafers, and the proportion of the age of the wife to that of the hufband. Let us then fuppofe, that every perfon in fuch fociety is of the fame age with his wife, and that, one with another, all the members when they enter may be reckoned 40 years of age, as many entering above this age as below it. It has been demonftrated by M. de Moivre and Mr Simpson, that the value of an annuity on the joint continuance of any two lives, fubtracted from the value of an annuity on the life in expectation, gives the true prefent value of annuity on what may happen to remain of the latter of the two lives after the other. (See ANNUITIES, Sec. I. Table II. and Sec. II. Tables, III, IV, and V.) In the prefent cafe, the value of the annuity to be enjoyed during the joint continuance of two lives, each 40, is by Table II. 9.826, according to the probabilities of life in the Table of Observations, formed by Dr Halley from the bills of mortality of Brellaw in Silefia. The value of a fingle life 40 years of age, as given by M. de Moivre, agree ably to the fame table, is 13'20; and the former fubtracted from the latter, leaves 337, or the true number of years purchase, which ought to be paid for any given annuity, to be enjoyed by a perfon 40 years of age, provided he furvives another perfon of the fame age, intereft being reckoned at four per cent per annum. The an nuity, therefore, being 30l. the prefent value of it is 30 multiplied by 337, or 101l. 25. IV. If, inftead of a fingle prefent payment, it is thought preferable to make annual payments during the marriage; what these annual payments ought to be is eafily determined by finding what annual payments during two joint lives of given ages are equivalent to the value of the reverfionary annuity in prefent money. Suppose, as before, that the joint lives are each 40, and the reverfionary annuity 30l. per annum. An annual payment during the continuance of two fuch lives is worth (according to Table II.) 9'82 years purchase. The annual payment ought to be fuch as, being multiplied by 9'82, will produce rol. the prefent value of the annuity in one payment. Divide then for by 982, and 103 the quotient will be the annual payment. This method of calculation fuppofes that the first annual payment is not to be made till the end of a year. If it is to be made immediately, the value of the joint lives will be increafed one year's purchafe; and, therefore, in order to find the annual payments required, the value of a prefent fingle payment must be divided by the value of the joint lives increafed by unity. If the fociety prefer paying part of the value in a prefent fingle payment on admiffion, and the reft in annual payments; and if they fix these annual payments at a particular fum, the prefent fingle payment paid on admiffion is found by fubtracting the value of the annual payment during the joint lives from the whole prefent value of the annuity in one payment. Suppose, for instance, the annual payments to be fixed at five guineas, the annuity to be sol. the rate of intereft four per cent, and the joint lives each 40: the value of the annuity in one prefent fingle payment is 101'il. The value of

5

five guineas or 5.25 per annum, is (5°25 multiplied by 982 the value of the joint lives) 51'55; which, fubtracted from ror'rl. gives 49's, the anfwer V. If a fociety takes in all the marriages amor perfons of a particular profeffion within a given district, and subjects them for perpetuity to a certain equal and common tax or annual payments, in order to provide life annuities for all the widows that shall refult from these marriages; fince, at the commencement of fuch an establishment, all the oldeft, as well as the youngest, marriages are to be entitled equally to the propofed benefit. a much greater number of annuitants will come immediately upon it than would come upon any fimilar eftablishment which limited itfelf in the admiffion of members to perfons not exceeding a given age. This will check that accumulation of money which fhould take place at firft, in order to produce an income equal to the difbursements at the time when the number of annuitants comes to a maximum; and therefore will be a particular burden upon the establishment in its infancy. For this fome compenfation must be provided; and the equitable method of providing it is, by levying fines at the beginning of the establishment on every member exceeding a given age, proportioned to the number of years which he has lived beyond that age. But if fuch fines cannot be levied, and, if every payment must be equal and common, whatever difparity there may be in the value of the expectations of different members, the fines muft be reduced to one common one, anfwering as nearly as poffible to the disadvantage, and payable by every member at the time when the eftablishment begins. After this, the establishment will be the fame with one that takes upon it all at the time they marry; and the tax or annual payment of every member adequate to its fupport will be the annual payment during marriage due from persons who marry at the mean age at which, upon an average, all marriages may be confidered as commencing. The fines to be paid at firft are, for every parti cular member, the fame with the difference between the value of the expectation to him at his prefent age, and what would have been its value to him had the scheme begun at the time he mar ried. Or, they are, for the whole body of members, the difference between the value of the common expectation, to perfons at the mean age of all married perfons taken together as they exift in the world, and to perfons at that age which is to be deemed their mean age when they marry. VI. Suppose we wish to know the prefent value of an annuity to be enjoyed by one life, for what may happen to remain of it beyond another life, after a given term; that is, provided both lives continue from the prefent time to the end of a given term of years; the method of calculating is this; Find the value of the aunuity for two lives, greater by the given term of years than the given lives; dilcount this value for the given terra; and then multiply by the probability, that the two given lives hall bath continue the given term and the product will be the anfwer. Thus, let the two lives be each 30, the term seven years, the annuity rol. interest for four per cent. given lives, increased by leven years, become each 37. The value of two joint lives, each 37, is (by

The

TABLE

creafed by unity, and twice the perpetuity the referved quotient, inftead of being fubtracted from twice the perpetuity, must be added to it, and the fum, not the difference, multiplied by the perpetuity increafed by unity. Example. Let the joint lives propofed be a female life aged 10, and a male life aged 15; and let the table of obfervations be the Sweden table for lives in general, and the rate of intereft 4 per cent. Twice the expectations of the two lives are 90'14 and 83°28. Twice the expectation of the oldef life, increafed by unity, and twice the perpetuity, is 134 28, which leffens by 90'14 (twice the expectation of the youngeft life), leaves 44'14 for the referved remainder. This remainder multiplied by 14 045 (the value of an annuity certain for 83.28 years), and the produ& divided by 83 28 (twice the expectation of the oldeft life), gives 12'744, the quo. tient to be referved; which fubtracted from double the perpetuity, and the remainder (or 371255) multiplied by the perpetuity increafed by unity (or by 26) gives 968 630, which divided by 9014 (twice the expectation of the youngest life) and the quotient fubtracted from the perpetuity, we have 14'254 for the required value. The value of an annuity certain, when the number of years is a whole number with a fraction added (as will be commonly the cafe) may be best computed in the following manner. In this example the number of years is 83 28. The value of an annuity cer tain for 83 years is 24'035. The fame value for 84 years is 24.072. The difference between thefe two values is o'37; which difference multiplied by 28 (the fractional part of the number of years), and the product ('0103) added to the leaft of the two values, will give 24'045 the value for 83°28 years. General Rule. Call the correct value (fuppofed to be computed for any rate of intereft) the first value. Call the value deduced (by the preceding problems) from the expectations at the fame rate of intereft, the fecond value. Call the value deduced from the expectations for any other rate of intereft the third value. Then the difference between the firft and fecond values added to or fubtracted from the third value, juft as the first is greater or lefs than the fecond, will be the value at the rate of intereft for which the third value has been deduced from the expectations. The following examples will make this perfectly plain. Example 1. In the two laft tables the correct values are given of two joint lives among mankind at large, without diftinguishing between males and females, according to the Sweden obfervations, reckoning the intereft at 4 per cent. Let it be required to find from thefe values the values at 3 per cent. and let the ages of the joint lives be fuppofed 10 and 1o. The correct value, table IV. (reckoning intereft at 4 per cent), is 16141. The expectation of a life aged rc is 45'07. The value deduced from this expectation at 4 per cent. by Prob. II. is 14'539. The value deduced by the fame problem from the fame expectation at 3 per cent. is 16 808. The difference between the first and fecond values is 1602, which, added to the third value (the first being greater than the second), makes 18'410, the value required. Example 11. Let the value be required of a ingle male life aged 10, at 3 per cent. intereft,

The

from the correct value at 4 per cent. according to the Sweden obfervations. First, or correct value, at 4 per cent. (by table III.) is 18 674. expectation of a male life aged to is 43'94The fecond value (or the value deduced from this expectation by problem I.) is 17838. The 3d value (or the value deduced from the fame expectation at 3 per cent.) is 21277. The difference between the first and fecond is 836; which (fince the first is greater than the fecond) must be added to the third; and the fum (that is, 22:113) will be the value required. The third value, at 5 per cent. is 15 286, and the difference added to 15 286 makes 161122 the value of a male life aged Io at 5 per cent according to the Sweden obfervations. The exact value at 5 per cent. is (by table III.) 16'014. Again: The difference between 16°014 (the correct value at 5 per cent.) and 15:286 (the value at the fame intereft deduced from the expectation), is 728; which, added (becaufe the firft value is greater than the tecond) to 13'335 (the value deduced at 6 per cent. from the expectation) gives 14063, the value of the fame life, reckoning intereft at 6 per cent. Thefe deductions, in the cafe of fingle lives particularly, are fo easy, and give the true values fo nearly, that it will be fcarcely ever neceflary to calculate the exact values (according to any given obfervations) for more than one rate of intereft. If, for inftance, the correct values are computed at 4 per cent. according to the obfervations, the values at 3, 34, 44 5. 6, 7 or 8 per cent. may be deduced from them by the preceding rules as occafion may require, without much labour or any danger of confiderable errors. The values thus deduced will feldom differ from the true values fo much as a tenth of a year's purchase. They will not generally differ more than a 20th or 30th of a year's purchase. In joint lives they will differ lefs than in fingle lives, and they will come equally near to one another whatever the rates of intereft are. The preceding tables furnish the means of determining the exact differences between the values of annuities, as they are made to depend on the furvivorship of any male or female lives; which hitherto has been a defideratum of confiderable confequence in the doctrine of life-annuities. What has made this of confequence is chiefly the multitude of focieties lately eftablished in this and foreign countries for providing annuities for widows. The general rule for calculating from thefe tables the value of fuch annuities is the following. Rule "Find in table III. the value of a female life at the age of the wife. From this value subtract the value in table IV. of the joint continuance of two lives at the ages of the husband and wife. The remainder will be the value in a fingle prefent payment of an annuity for the life of the wife should the be left a widow. And this laft value divided by the value of the joint lives increafed by unity, will be the value of the fame annuity in annual payments during the joint lives, and to commence immediately." Example. Let the age of the wife be 14, and of the husband 30. The value in table III. (reckoning intereft at 4 per cent.) of a female life aged 24, is 17'252. The value in table IV. of two joint aves aged 24 and 30, is 137455;

co, bounds the above province, to which it gives name, on the S. and falls into the Atlantic Ocean, near Santa Cruz.

which fubtracted from 17'252 leaves 3'797, the value in a tingle prefent payment of an anquity of 11. for the life of the wife after the hufband; that is, for the life of the widow. The annuity, therefore, beng fuppofed 201. its value in a fingle pay ment is 20 multiplied by 3797, that is, 75 941. And this laft value divided by 14'455 (that is, by the value of the joint lives increated by unity), gives 515, the value in annual payments beginning immediately, and to be continued du. ring the joint lives of an annuity of 20 1. to a wife aged 24 for her life, after her husband aged 30.

SURUNGA, or SUISJU, a fea-port town of Japan, in the ifle of Niphon, and capital of a province fo named; 170 miles E. of Meaco.

SURWILISKI, a town of Lithuania, in the late palatinate of Wilna, now annexed to Pruffia: 36 miles SSE. of Wilna.

(1.) SURY, or SURY EN VAUX, a town of France, in the dep. of the Cher: 3 miles N. of Sancerre, and 12 NE. of Henrichmont.

(2.) SURY LE COMTAL, a town of France, in the dep. of the Rhone and Loire : 6 miles SE. of Montbrison, and 12 NW. of St Etienne. Lon. 21. 51. E. Ferro. Lat. 45. 32- N.

SURYA, the orb of the fun perfonified and adored by a fect of Hindoos as a god, like the PHOEBUS of Greece and Rome. The fect who pay him particular adoration are called Sauras. Their poets and painters defcribe his car as drawn by feven green horfes, proceeded by Arun, or the Dawn, who acts as his charioteer, and followed by thoufands of genii worshipping him and modulating his praifes. He has a multitude of names, and among them 12 epithets or titles, which denote his diftinct powers in each of the 12 months; and he is believed to have descended frequently from his car in a human shape, and to have left a race on earth, who are equally renowned in the Indian ftories with the Heliadai of Greece: it is very fingular, that his two fons called Asinau and Aswinicumarau, in the dual, fhould be confidered as twin-brothers, and paint-, ed like Caftor and Pollux; but they have each the character of Efculapius among the gods, and are believed to have been born of a nymph, who, in the form of a mare, was impregnated with funbeams.

SURZEE, a town of Switzerland in Lucern, near lake Sempach, 5 miles S. of Lucern. SURZUR, a town of France, in the departIment of Morbihan; 6 miles SE. of Vannes.

(I.) SUS, or Suz, or Sous, in geography, a province of Africa, belonging to Morocco; bounded on the N. by mount Atlas; E. by Gefula; S. by the fands of Numidia and Sus; and W. by the Ocean. It is a vel fertile country, abounding in corn, fugar canes, dates, &c. In the mountains the inhabitants are free and governed by their own chiefs, or fcheiks. TARODANT is the capital. (Brookes.) But the rev. C. Cruttwell makes Darah its boundary on the E. and Nun on the S. He adds, that Aguadir, ToMA, TECENT, and Messa, are alfo among its chief towns.

(II) Sus, Suz, or Sous, a river of Africa, which rifes from Mount Atlas, in the empire of Moroc

(II.) Sus, the HOG, in zoology, a genus of quatrupeds belonging to the class of mammalia and order of bellua. There are four cutting teeth in the upper jaw, whofe points converge; and, for the most part, fix in the lower jaw, which ftand forwards. There are two tulks in each jaw, thofe in the upper jaw being short, while thofe of the under jaw are long, and extend cut of the mouth. The fnout is prominent, moveable, and has the appearance of having been cut off, or truncated. The feet are armed with divided or cloven hoofs. There are fix fpecies:

I. SUS ETHIOPICUS, the Engallo, or Ethiopian bog, has no fore teeth, but Imail tusks in the lower jaw, very large ones in the upper, in old boars bending towards the forehead in form of a feinicircle: nofe broad, depreffed, and almost of a horny hardness: head very large and broad: beneath each eye a hollow, formed of loole skin, very foft and wrinkled; under these a great lobe or wattle, lying aimoft horizontal, broad, fat, and rounded at the end, placed fo as to intercept the view of any thing below from the animal. Between thefe and the mouth on each fide, there 18 a hard callous protuberance. The mouth is fmail: skin dulky: brittles difpofed in fafciculi, of about five each; longeft between the ears and on the beginning of the back, thinly di perled on the reft of the back. Ears large and fharp pointed, infide lined with long whitith hairs: tail flender and flat, not reaching lower than the thighs, and is covered with hairs difpofed in fafciculi. Body longer, and legs fhorter, than in the common fwine: its whole length 4 feet inches; height before, a feet 2 inches: but in a wild ftate, it grows to an enormous fize. These animals inhabit the hotteft parts of Africa, from Senegal to Congo, alfo the island of Madagascar. They are very fierce, active and swift; they will not breed with the domeftic fow. They burrow under ground, into which they dig with furpiling expedition. Buffon confounds this with the Afri can.

2. SUS AFRICANUS, the African hog, has only 2 fore teeth in the upper jaw, and 6 in the lower; on each lide 6 grinders. The body is covered with very long and fine brifties; the tail reaches to the first joint of the hind leg; is very flender, and ends in a large tuft: the head is long, Lofe flender; the upper jaw extending far beyond the lower; the ears narrow, erect and pointed, with very long briftles at the end. They inhabit Airica, from Cape Verd to the Cape of Good Hope.

3. SUS BABYRUSSA, the Indian beg, has 4 cutting teeth in the upper, lix in the lower jaw; ten grinders to each jaw; in the lower jaw two talks pointing towards the eyes, and standing near 8 inches out of their fockets; from two fockets on the outfide of the upper jaw two other teeth, 12 inches long, bending like horus, their ends almoft touching the forehead: ears Imall, erect, fharp-pointed along the back are tome weak briftles; on the rest of the body only a fort of wool, fuch as is on lambs: the tail long, ends in a tuft, and is often twisted: the body plump and

B 2

iquare.