Encyclopædia Britannica: Or, A Dictionary of Arts, Sciences, and ...

SURVEYING, the art of measuring land; that is, of taking the dimenfions of any tract of ground, laying down the fame in a map or draught, and finding the content or area therof. See GEOMETRY.

SURVEYOR, a person who has the oversight and care of confiderable works, lands, or the like.

SURVEYOR, likewife denotes a gauger; as also a person who furveys lands, and makes maps of them.

SURVIVOR, in law, fignifies the longeft liver of joint tenants, or of any two perfons jointly interested in a thing. SURVIVORSHIP, is that branch of mathematics which treats of reverfions payable provided one or more particular perfons furvive certain others. By reverfions are meant payments not to take place till fome future period. Survivor fhip forms one of the most difficult and complicated parts of the doctrine of reverfions and life-annuities. It has been very fully treated of by Mr Thomas Simpfon in his Select Exercifes; and brought to a state of very great perfection by Dr Price and Mr Morgan, who have beftowed a great deal of attention on this subject.

The calculations are founded on the expectation of lives at different ages, deduced from tables formed from bills of mortality, of which fee several examples under the article Bills of MORTALITY. By the expectation of life is meant the mean time that any fingle or joint lives at a given age is found to continue; that is, the number of years which, taking one with another, they actually enjoy, and may be confidered as fure of enjoying; thofe who furvive that period enjoying as much more time in proportion to their number as those who fall short of it enjoy lefs. Thus, fuppofing 46 perfons alive all 40 years of age, and that one will die every year till they are all dead in 46 years, half 46 or 23 will be the expectation of each of them. If M. de Moivre's hypothefs were true, that men always decrease in an arithmetical progreffion, the expectation of a fingle life is always half its complement (A), and the expectation of two joint lives onethird of their common complement. Thus, fuppofing a man 40, his expectation would be 23, the half of 46, his complement; the expectation of two joint lives, each 40, would be 15 years 4 months, or the third part of 46.

The number expreffing the expectation, multiplied by the number of fingle or joint lives (of which it is the expectation), added annually to a fociety, gives the whole number living together, to which fuch an annual addition would in time grow. Thus, fince 19, or the third of 57, is the expectation of two joint lives, whofe common age is 29, twenty marriages every year between perfons of this age would in 57 years grow to 20 times, 19, or 380 marriages, always exifting together. And fince the expectation of a fingle life is always half its complement, in 57 years 20 fingle perfons added annually to a town will increase to 20 times 28.5, or 570; and when arrived at this number, the deaths every year will just equal the acceffions, and no farther increase be poffible. It appears from hence, that the particular proportion that becomes extinct every year, out of the whole number con. ftantly exifting together of fingle or joint lives, muft, where ver this number undergoes no variation, be exactly the fame with the expectation of thofe lives, at the time when their existence commenced. Thus, was it found that a 19th part of all the marriages among any bodies of men, whofe VOL. XVIII. Part I.

ship.

numbers do not vary, are diffolved every year by the deaths Survivor of either the hufband or wife, it would appear that 19 was, at the time they were contracted, the expectation of these marriages. In like manner, was it found in a fociety, limited to a fixed number of members, that a 28th part dies an nually out of the whole number of members, it would ap pear that 28 was their common expectation of life at the time they entered. So likewise, were it found in any town or district, where the number of births and burials are equal, that a 20th or 30th part of the inhabitants die annually, it would appear that 20 or 30 was the expectation of a child juft born in that town or diftrict. Thefe expectations, therefore, for all fingle lives, are eafily found by a table of obfervations, fhowing the number that die annually at all ages out of a given number alive at thofe ages; and the general rule for this purpofe is, to divide the sum of all the living in the table, at the age whofe expectation is required, and at all greater ages, by the fum of all that die annually at that age and above it; or, which is the fame, by the number (in the Table) of the living at that age; and half unity fubtracted from the quotient will be the required expectation. Thus, in Dr Halley's table, given in the article ANNUITY, 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 subtracted from the quotient, gives 34.15 for the expectation of 20.

In calculating the value or expectation of joint lives, Mr de Moivre had recourse to the hypothefis, that the probabilities of life decrease in a geometrical progreffion; believing that the values of joint lives, obtained by rules derived from it, would not deviate much from the truth. But in this he was greatly mistaken; they generally give refults which are near a quarter of the true value too great in finding the prefent value of one life after it has furvived another in a fingle payment, and about ths too great when the value the value is fought in annual payments during the joint lives. They ought therefore to be calculated upon the hypothefis (if they are calculated on hypothefis at all), that the probabilities of life decrease in arithmetical progreffion, which is not very far from the truth. Even this hypothefis never correfponds with the fact in the first and last periods of life, and in fome fituations not in any period of life. Dr Price and Mr Morgan therefore have given tables of the value of lives, not founded on any hypothefis, but deduced from bills of mortality themselves. Some of these we shall give at the end of this article. Mr Morgan has likewife given rules for calculating values of lives in this manner.

M. de Moivre has alfo fallen into mistakes in his rules for calculating the value of reverfions depending on furvivorfhip: thefe have been pointed out by Dr Price in the third effay in the firft volume of his Treatife on Reverfionary Payments; who has alfo given proper rules for calculating thefe values, the moft important of which are comprehended in the following paragraphs.

Suppose a fet of married men to enter into a fociety in Method of order to provide annuites for their widows, and that it is finding the limited to a certain number of members, and conftantly kept uitst number of up to that number by the admiffion of new members as the that will old ones are loft; it is of importance, in the firft place, to come on a know the number of annuitants that after fome time will fociety. come upon the eftablishment. 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 fhall die firft; it is evident, that the B b number

(A) By the complement of a life is meant what it wants of 86, which M. de Moivre makes the boundary of human life. Thus if a man be 30, the complement of his life is 56.

ages.

and to a maximum, in 30 years, fuppofing, with M. de Survivor Moivre, 86 to be the utmoft extent of life. The fame will ship. happen to the fecond clafs in 40 years, and to the third in 50 years. But the whole body compofed of these claffes will not come to a maximum till the fame happens to the fourth or youngest class; that is, not till the end of 60 years. After this the affairs of the fociety will become stationary, and the number of annuitants upon it of all ages will keep always nearly the same.

What a

man ought

to pay in a

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 on 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 intitle his widow to a certain annuity for her life, let us fuppofe the annuity 3 fingle payper annum, and the rate of intereft four per cent. It is evi-ment to dent, that the value of fuch an expectation is different, ac- entitle his cording to the different ages of the purchafers, and the widow to a proportion of the age of the wife to that of the husband. certain an Let us then fuppofe that every person in fuch a 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 present value of annuity on what may happen to remain of the latter of the two lives after the other.

Survivor- number of widows that have ever exifted in the world, fhip. would in this cafe be equal to half the number of marriAnd what would take place in the world must alfo, on the fame fuppofitions, take place in this fociety. In other words, every other perfon in fuch a fociety leaving a widow, there must 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 set of such 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 most towns the decrease in the probabilities of lifeis in fact nearly uniform. According to the Breslaw Table of Observation (fee ANNUITY), 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 hypothetis is inconfiderable; and its effect, in the prefent cafe, is to render the duration of furvivorship longer than it would otherwife 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 decrease 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 fomewhat greater than half the number of the fociety; but, according to the London Table, a good deal greater. This, however, has been detertermined 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 hufbands are generally older than wives, and males have been found to die fooner than females, as appears inconteftably from feveral of the tables in Dr Price's Treatife on Reverfions. It is therefore more than an equal chance that the husband will die before his wife. This will increase confiderably the duration of survivorship 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.

When the

airives at its maxi. mum.

If the fociety comprehends in it from the firft all the number of married people of all ages in any town, or among any clafs annuitants of people where the numbers always continue the fame, the whole collective body of members will be at their greateft age at the time of the establishment 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 continually 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 greatest poffible number of the oldeft widows; and therefore not till there has been time for an acceffion to the oldeft 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 fome 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 firft clafs, will come to their greateft age,

In the present cafe, the value of an 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 Breflaw in Silefia. The value of a fingle life 40 years of age, as given by M. de Moivre, agreeably to the fame table, is 13.20; and the former fubtracted from the latter, leaves 3.37, 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 annuity, therefore, being 301. the prefent value of it is 30 multiplied by 3.37, or 101l. 2s.

num.

nuity.

in an

If, instead of a fingle prefent payment, it is thought pre- What he ferable to make annual payments during the marriage; ought to what these annual payments ought to be is eafily determi-nual pay ned by finding what annual payments during two joint lives ments. of given ages are equivalent to the value of the reverfionary annuity in prefent money. Suppofe, as before, that the joint lives are each 40, and the reverfionary annuity 301. per anAn annual payment during the continuance of two fuch lives is worth (according to Table II.) 9.82 years pur. chafe. The annual payment ought to be fuch as, being multiplied by 9.82, will produce 101.11. the present value of the annuity in one payment. Divide then 101.1 by 9.82, and 10.3 the quotient will be the annual payment.. This method of calculation supposes 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 increased one year's purchase; 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.

thip.

lives be each 30, the term feven years, the annuity 1. ro, Survivorintereft four fer cent. The given lives, increafed 'by leven years, become each 37. The value of two joint lives, cach 37, is (by Table II.) 10.25. The value of a fingle lite at 37 is (by the table under the article ANNUITY) 13.67. The former fubtraced from the latter is 3.42, or the value of an annuity for the life of a perfon 37 years of age, after another of the fame age, as has been shown above, 3.42 dif counted for feven years (that is, multiplied by 0.76 the value of 11. due at the end of feven years) is 2.6. The probability that a fingle life at 30 fhall continue feven years is 2 (B). The probability, therefore, that two fuch lives fhall continue feven years, is 474, or in decimals 0.765; and 2.6 multiplied by 0.765 is 1.989, the number of years purchase which ought to be given for an annuity to be enjoyed by a life now 30 years of age, after a life of the fame age, provided both continue feven years. The annuity then being 10l. its prefent value is 1. 19.89.

gurvivor, at a particular fum, the prefent fingle payment paid on admiffion is found by fubtracting the value of the annual pay ment during the joint lives from the whole prefent value of the annuity in one payment. Suppose, for inftance, the annual payments to be fixed at five guineas, the annuity to be 301. the rate of intereft four per cent. and the joint lives each 40; the value of the annuity in one prefent fingle payment is 101.11. The value of five guineas or 5.25 per annum, (5.25 multiplied by 9.82 the value of the joint lives) 51.55; which, fubtracted from 101.11. gives 1, 49.5, the answer. If a fociety takes in all the marriages among perfons of a particular profeffion within a given district, and fubjects 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 youngeft, marriages are to be intitled equally to the proposed benefit, a much greater number of annui. tants will come immediately upon it than would come upon any fimilar establishment which limited itself in the ad miffion of members to perfons not exceeding a given age. This will check that accumulation of money which fhould take place at first, in order to produce an income equal to the disbursements 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 compensation 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 must be reduced to one commor one, answering as nearly as poffible to the disadvantage, and payable by every member at the time when the efta. blishment 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 perfons 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 particular 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 married. 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. Suppose we wish to know the prefent value of an annuity Method of to be enjoyed by one life, for what may happen to remain finding the of it beyond another life, after a given term; that is, proprefent value of an vided both lives continue from the present time to the end annuity ro of a given term of years; the method of calculating is this: he enjoyed by one life Find the value of the annuity for two lives, greater by the after the given term of years than the given lives; difcount this value expiration for the given term; and then multiply by the probability, that the two given lives fhall both continue the given term; and the product will be the answer. Thus, let the two

of ano

ther.

value of an

what may

after another, pro

continues

Suppose the value is required of an annuity to be enjoyed Method of for what may happen to remain of one life after another, pro- finding the vided the life in expectation continues a given time. Find the prefent value of the annuity for the remainder of the annuity for life in expectation after the given time, which is done in this happen to manner: Multiply the prefent value of the life at the given remain of time by the prefent value of 11. to be received at that time, one life and multiply the product again by the probability that the life in expectation will continue fo long. Let the given time vided the which the life in expectation is to continue be 15 years, and life in exlet the perfon then be arrived at 50 years of age. A life pectation at fifty, according to M. de Moivre's valuation of lives, and a given reckoning intereft at four per cent. is worth 11.34 years term. purchase. The prefent value of 1 1. to be received at the end of 15 years, is 0.5553, and the probability that a life at 35 will continue 15 years is 146. These three values multiplied into one another give L. 4.44 for the prefent value of the life in expectation. 2. Find the value of the reverfion, provided both lives continue the given time, by the rule given in parag. 5th. 3. Add these values together, and the fum will be the answer in a fingle prefent payment. We shall now illuftrate this rule by an example.

490

An annuity of 101. for the life of a perfon now 30, is to commence at the end of 1 years, if another perfon now 40 fhould be then dead; or, if this fhould not happen at the end of any year beyond 11 years in which the former fhall happen to furvive the latter: What is the present value of fuch an annuity, reckoning intereft at four per cent. and taking the probabilities of life as they are in Dr Halley's table, given in the article MORTALITY?

3 5

The value of 101. per annum, for the remainder of the life of a perfon now 30, after 11 years is L. 69.43. The probability that a person 40 years of age fhall live 11 years, is, by Dr Halley's table, 1. The probability, therefore, that he will die in 11 years, is fubtracted from unity (c), or 1; which multiplied by 1. 69.43, gives 1. 17.16.--The value of the reverfion, provided both live 11 years, is 171. and this value added to the former, makes 1. 34.16. the value required in a fingle present payment; which payment divided by l. 11.43, the value of two joint lives, aged 30 and 40, with unity added, gives 31.; or the value required in annual payments during the joint lives, the firft payment to be made immediately. Bb 2 TABLE

(B) The probability that a given life fhall continue any number of years, or reach a given age, is (as is well known) the fraction, whofe numerator is the number of the living in any table of observations oppofite to the given age, and denominator, the number oppofite to the present age of the given life.

(c) For the difference between unity and the fraction expreffing the probability that an event will happen, gives the probability that it will not happen.

ship. TABLE I. Showing the Prefent Values of an Annuity of L. on a Single Life, according to M. de Moivre's Hypothefis.