68 aged 90 and upwards, and the remaining I was living aged 101. This was David Rennie, who was baptized 28th Feb. 1755, and died 2nd March, 1857, in his 103rd year. Some add. facts will be given under head of TONTINES. ANNUITIES.-An annuity has, in the strict technical acceptation of the word, been defined by Lord Coke to be "a yearly payment of a certain sum of money granted to another in fee, for life or years, charging the person of the grantor only." If the annuity be secured solely out of the rent of land, it is called a Rent-charge. [RENT-CHARGE.]

An annuity may be receivable during a definite number of years, in which it is termed an annuity certain. The value of annuities of this kind depends only on the rate of int. for money, and the number of years during which the annuity is payable. If the annuity be payable for ever, it is called a perpetual annuity or a perpetuity. If for a limited number of years, a term annuity. If not till after a certain date, it is called a deferred annuity. If its payment be dependent on the happening or non-happening of some particular event, it is called a contingent annuity. If the commencement or the continuance of the annuity is any way dependent upon the duration of any life or lives, it is called a LIFE ANNUITY. It is with these we have especially to deal in these pages.

A Life annuity may partake of some of the incidents of an annuity certain. If the annuity be for the whole period of any particular life, it is called a whole-term annuity. If it be on a given life for a certain number of years, it is called a term annuity; for a few years only a short-term life annuity. If the annuity be payable on any one life, provided another be then living, or providing some event happen or do not happen, it is a contingent life annuity. If it is not to be entered upon till after the death of some person or persons previously living, it is called a reversionary annuity. When it is continued only for a term of years, provided an individual or individuals then living shall survive that term, it is called a temporary life annuity. When it is payable during the lives of several persons living at the same time, it is called a joint-life annuity. When to the survivor of several lives, then a survivorship annuity. There are also increasing and decreasing annu. These may depend upon life or otherwise.

There is yet another definition. An annuity is curtate when it ceases with the last payment prior to the death of any specified life: complete when the payment is to be continued up to the day of the annuitant's death. See ANNUITY APPORTIONMENT ACT. The values of a life annuity must necessarily vary with the contingencies upon which it is dependent. There are actuarial rules for dealing with all the cases named. Two or more of the preceding contingencies may be combined; but however complex, they fall within the grasp of the experienced actuary.

ANNUITIES ON LIVES, HIST. OF.-The hist. of life annu. has remained to be written. We propose here to supply a considerable contribution towards the materials required for the purpose. By way of explaining the plan of the present chap.—which must of necessity be one of the most extended in this work-we may at once say that our treatment of life annu. will be historical, and not scientific. It would indeed be a matter of much interest to the actuary that the progressive formulæ evolved in the scientific treatment of the doctrine of life annu. should be recorded. We leave that task for other hands: we have ample work before us. It seems important further to remark, that while, at the present period, life annu. transactions are so far secondary and subordinate to life ins. as not prob. to bear the proportion of one-tenth-taking the cap. sunk in the purchase of life annu. as against the cap. to be realized by the payment of claims under life pol.—yet the facts were formerly very much the other way. For several centuries before the present life annu. constituted a recognized mode of investment-life ins. was regarded as a speculation. This will to a considerable extent account for the fact, presently to be noted, that all the earlier writers upon life contingencies directed their attention to annu. and life ins. was an afterthought. Or in the apt dictum of Mr. E. J. Farren: "As life assu. formule were eventually derived from annu. calculations, so it was not until an after-period that life assu. attracted notice as a distinct subject."

It is impossible to fix at what period of the world's history annuities on lives were first brought into practice. It is only required for our present purpose to know when they first became subjected to any measure of value. Most of the best-informed writers agree that the first judicial occasion for valuing annu. on lives arose in consequence of the Falcidian Law (Lex Falcidia de Legatio), which in B.C. 40 was adopted in the Roman empire; and which declared that a testator should not give more than three-fourths of his property in legacies, and thus one-fourth was required to be secured to his legal heirs. It became necessary, in the execution of this law, to value all such legacies as were charged upon the succession for limited terms, or as annuities for life. It is possible that several means of accomplishing this were attempted. We learn from Æmilius Macer (A.D. 230), that the method which had been in common use at that period was as follows: From the earliest age until 30, to compute 30 years; from age 30, so many years are computed as are wanting until 60: therefore never more than 30 years are computed.

It was no doubt the employment of this defective method which caused the great jurist, the Prætorian Præfect Úlpianus, to give his attention to the subject. It is impossible to ascertain what materials he called to his aid in preparing his estimate, nor is it our present purpose to do so. All that can be learned on that subject we shall present to

the reader under the head of ROMAN LIFE TABLE. We only now present the results, reduced from their orig. crude form into a tabular arrangment, as follows:

It seems pretty certain that the element of interest was not taken into account; but this we shall discuss hereafter. The table is here presented as the first known measure of life annuity values graduated with reference to age. This very table has been brought into use during the present century by the Tuscan government for the valuation of life annuities!

By way of preface to the hist. of life annuities in England it seems necessary to call attention to the fact that in earlier times all int. for the use of money was absolutely forbidden: first, by the canons of the Church, and afterwards by direct legal enactment; while in many parts of the continent of Europe, where the power of the Church was more despotic, the prohibition was even more severe. The first real enactment against usury in Gt. Brit. (for the distinction between usury and interest had not then been created) was in the reign of Richard I. A.D. 1197, forbidding Christians to take any recompense for money lent. Other similar enactments followed. USURY.] The Jews do not appear to have been prohibited in the enactment of Richard; and they and the Lombards became the great money-lenders of Europe.

Age.

Years' Purchase.

Birth to 20

30

20 25

28

25,, 30

25

30,, 35

22

35,, 40

20

40,, 41

19

41,, 42

18

42,

43

17

In 1282 the king of Scotland gave Eric, king of Norway a marriage portion of 14,000 marks with his daughter; reserving to himself an option of giving a life-rent of lands of the ann. value of 700 marks as an equivalent for half that sum. The annu. on the life of Margaret, then in her 21st year, was thus valued at 10 years' purchase. Whether this was the rule of the period, or merely an accident, cannot now be determined.

As the law did not recognize the lending of money at usance, it could not impose any limits upon the rate of int. charged. But the Jews and Lombards carried on the bus., charging no doubt most extortionately-as indeed they must, when the whole principal was at stake by reason of the transaction being in violation of the law. Hence, no doubt, was passed in 1545, 9th Henry VIII., c. 9, an act restricting the rate of interest to 10 p.c. INTEREST OF MONEY.]

The celebrated Benvenuto Cellini, in his biography, relates that in the year 1552, he travelled to Rome to see his banker, Bindo Altoviti, who, as was not unusual at that period, was learned and a protector of the arts and sciences. His fortune had suffered by the political intrigues of the times, and instead of paying Cellini 1200 gold thalers due to him, it was arranged between them that he should pay an annuity of 15 p.c. during the life of Cellini in lieu of the money. Cellini speaks of other similar contracts, and from the manner in which he speaks of them it would appear that such transactions were not new, but long known, and of common occurrence.

In a work pub. in 1554, by Dr. Thomas Wilson, entitled A Discourse upon Usurie, etc., we find the following:

A corporation taketh a 100 li. (£) of a man to give him 8 in the roo li. during his life, without restitution of the principall. It is no usurie, for that here is no lending, but a sale for ever of so much rent for so much monie. Likewise it is, if a private man hath a thousand pound lieing by him, and demandeth for his life and his wive's life, a 100 li. by the yeere, and never to demand the principall. It is a bargaine and sale, and no usurie, for that the principall is not to be restored againe at anie time. And therefore no lending can be pre-supposed.

Here is shown a clear perception of the business of life annuities, but shaped in such a form as to avoid the usury laws.

It appears to us, after a careful consideration of the subject, that the system of granting loans, repayable by annuities on lives, had been specially devised with a view to the evasion of the laws against usury prior to 1545; and that the evasions continued in practice after the passing of that Act. The authorities we shall have occasion to quote as we proceed will, we think, fully sustain this view.

The Act of 1545 had been repealed in 1552; but in 1570 the legal rate of int. was again fixed at 10 pc.

During the greater part of the 16th century probably much speculation was carried on in annuities by private dealers; especially in the latter half. The annals of that period teem with records of the doings of one Audley, who, although originally only a poor clerk with six shillings a week, was so neat an adept in the tricks of law, and so keen in his annuity dealings, which chiefly consisted in purchasing annuities well secured upon property, that he became one of the richest men of his time. His mode of action may be best judged of by the reply made to one of his victims, who accuses him of having no conscience," We monied people must balance accounts. If you don't pay me my annuity, you cheat me; if you do, I cheat you."-Francis.

Malynes, an English merchant, in his famous Lex Mercatoria, pub. 1622, gives the following remarkable account of the annuity dealings of that period :

Hauing intreated of monies deliuered at interest without casualtie, and so termed Vsurie by reason

of the contract of benefit without aduenture: it is conuenient to handle of monies deliuered vpon Liues, when merchants doe giue twelue vpon the hundred without pawne, called beyond the seas after the pennie 8: the moitie whereof with pawne is six upon the hundreth, or double eight, according to the penny 16 as aforesaid. The pennie 8 is 124, for eight times 12 maketh one hundreth: so the pennie 16 which is vsed for rents vpon houses or lands is 6, according to which pennie 8 vpon one life or double for one yeare (so they all liue) is equall vpon eight liues. This is much used in diuers cities beyond the seas, to draw monies into their hands, as for example:

One hundreth pounds is deliuered to haue two hundreth pounds for it at the yeares end vpon eight liues, if they all liue, you haue two hundreth pounds to buy a perpetuall rent, or sometimes as it hath beene, to giue 20 per 100 for the yeare, and so from yeare to yeare, and dying the principall is theirs. One hundreth pounds for eight liues (by equalitie decreasing the pennie eight) is £12 10 for two liues, £11 2 for three liues, £10 for foure liues, £9 1 9 pence, is £6 5 for eight liues. The citie of Amsterday (Amsterdam) was wont to giue good consideration, and did obserue this order vpon a hundreth guelders:

For eight liues gaue...

Of nine liues

Vpon ten liues they gaue
Vpon eleuen liues

..£16 13 4

19 I 0
22
...
4 5
26 13 4

Vpon twelue liues
Vpon seuen liues
Vpon six liues

£33 6 8 14 6 4

13 6 8

Here you are to remember the obseruations of assurors, whether the persons be young or old, sober in their diet and behauiour, much travelling abroad, or staying at home, subject to sickenesse, and the like considerations.

Monies deliuered vpon annuities for rents, seeme more reasonable than pensions vpon liues, because you beare onely the aduenture of the decay of houses or destruction of them in time of warre: and much more should bee giuen without pawne, than pawne or morgage.

It is commodious for a man, hauing waste grounds and intending to build vpon them, to take much money vpon rent after 6 pro cent. which many cities giue continually to increase their wealth and inhabitants. And because the valuation of their money doth often alter and is inhaunced, whereby all things become deerer, the parties are aduised to haue their rents paid in specie, in Crownes, Dollers, Ducats to be paied as formerly they were currant; the dangers in times of wars causeth rents to decrease, for the ancient rent is alwaies first to be paid, although all others be losers.

In 1624 the legal rate of int. in England was reduced from 10 to 8 per cent.; and remained at this latter rate until 1651.

It seems doubtful whether any tables for valuing annuities existed before the early part of the 17th century. About 1620, William Webster published his tables of simple int. and also "His Tables of Compound Int., with true Valuations of Annuities, Leases, Fines, and Reversions,” 2nd ed. 1629. These tables, however, and the values deduced from them, are all for fixed terms of years, and the contingency of life is not even contemplated by them. The same may be said of the tables of Wm. Purser, pub. 1634: Compound Int. and Annuities, containing the art of Decimal Arithmetic; also the Value of Annuities.

In 1651 the Rump Parliament reduced the legal rate of int. from 8 to 6 p.c. This was legitimatized in 1660; and it remained at that rate until 1714.

In 1658 John Newton pub. Table of the Value of Annuities; but again there is no mention of Life Annuities.

In 1661 H. James Hodder's famous book of arithmetic was first pub., and therein was "A very brief and necessary table to find out the present worth of the annuity, or yearly rent for 21 years or under, after the rate of 6 p.c. p.a."

In 1669 a "second impression" of Roger Clavell's Tabula Faneratoria was pub., in which was contained: Tables for the forbearance, discompt, and purchase of annuities to 31 years, at the rate of 6 p.c. p.a., according to the late Act of Parl. Still not a word on the subject of Life Annuities!

In the middle of the 17th century, Pascal, Fermat, and Huygens laid the foundation of the mathematical theory of probabilities, which even in its first crude stages was of great service in the investigation of life contingencies; and about the same time John Wallis, an Englishman, pub. a treatise on Algebra, which also had to play an important part in the practical development of the last-named science. Huygens' tract, which appeared in 1658, was the first systematic treatise on the Doctrine of Chances which had been pub. : the doctrine of chances taught us how to estimate life contingencies: the science of life contingencies enabled us to measure the value of life annuities.

The mention of some of the preceding names carries our minds at once to the continent of Europe, whither we must pursue the subject, well assured that our enterprise will bring an abundant reward.

On the 25th April, 1671, it was resolved by the States General of Holland to negociate funds through the medium of Life Annuities. On the 30th July following the resolution was confirmed; and on that same day was presented to the Assembly a report by the Grand Pensionary, John De Wit, explaining the basis upon which such an enterprise could be successfully carried out. This event constitutes an important epoch in the hist. of Life Annuities. We have, however, no time to dwell upon the circumstance: we must approach the report itself:

Noble and Mighty Lords.-In so extensive an administration as that of the united country of Holland and West Friesland, it is better, as I have several times stated to your lordships, for several reasons perfectly well known to you, to negociate funds by Life Annuities, which from their nature are infallibly terminable, than to obtain them at int. which is perpetual, or by redeemable annuities; and that it is likewise more useful for private families who understand economy well, and know how to make a good employment of their surplus in augmenting their capital, to improve their money by life annuities than to invest it in redeemable annuities, or at int. at the rate of 4 p.c. p.a.; because the above-mentioned life annuities, which are sold even at the present time at 14 years' purchase, pay in fact much more in proportion than redeemable annuities at 25 years' purchase. I have consequently respectfully to submit

to your lordships the unchallengeable proof of my assertions, and at the same time to respond to the wish manifested by the members of this body to have such proof in writing. That proof, founded on a solid basis, is proposed to your High Mightinesses in the following manner :

Value of Life Annuities in proportion to Redeemable Annuities.

I lay down the following presuppositions in order to determine the proportion of a life annuity to a redeemable annuity. For example, in presupposing that the redeemable annuity is and will be current at 25 years' purchase, or at the rate of 4 p.c. p.a., we must find at how many years' purchase the life annuity should be sold, to be in proportion to the aforesaid redeemable annuity, in such manner that the life annuity may, if not with mathematical precision, at least in its discovered value, be more advantageous to the purchaser than an annuity redeemable with the same capital.

FIRST PRESUPPOSITION.

I presuppose that the real value of certain expectations or chances of objects of different value, must be estimated by that which we can obtain from equal expectations or chances dependent on one of several equal contracts. Let us take, for example, a small matter, and under circumstances intelligible at first sight: A person has two different expectations or chances, which may easily lead, the one to nothing, the other to 20 stuyvers. If, by one or several equal contracts, he can obtain for 10 stuyvers two like expectations or chances, we must estimate that the two aforesaid chances are worth to him exactly 10 stuyvers, because he can really obtain for 10 stuyvers these two expectations or chances by making an agreement with another person that each of them shall take 10 stuyvers, and then gamble or draw lots, by odd or even, head or tail, blank or prize, or in some such way, to determine which of the two should have the 20 stuyvers; thus by the said contract, equal in every regard, he evidently finds himself in the position of having in reality the two expectations or chances, the one of nothing, the other of 20 stuyvers.

SECOND PRESUPPOSITION.

That in taking at pleasure some years of a man's life, limited to the time when he is in his full vigour, and neither too young nor too advanced in age (this space of time shall be here 50 years, namely, from the third or fourth year of his age up to the fifty-third or fifty-fourth year), it is not more likely that this man should die in the first half-year of a given year than in the second half: similarly, it is not more likely that he should die in the second half-year of the aforesaid year than in the first half. But although it depends entirely on chance whether this man, after having lived to the given year, and dying in the course of that year, should demise in its first or second half, one finds nevertheless, in this regard, an equality of likelihood or chance similar to the case of a tossed penny, where there is an absolute equality of likelihood or chance that it will fall head or tail, although it depends entirely upon chance as to the side on which it shall turn, and this to so high a degree that the penny may fall head 10, 20, or more times following without once falling tail, or vice versa.

THIRD PRESUPPOSITION.

That a man having passed the aforesaid vigorous time of his life, namely, the fifty-third or fifty-fourth year of his age, it begins to be more likely that he should die in a given year or half-year of the second period than has previously been the case; or that it is not likely, with respect to another man of like constitution or state of body, that the latter should die in less than a year or half-year of the said vigorous time of his life; whilst this likelihood or chance of dying in a given year or half-year of the ten first following years, namely, from 53 to 63 years of his age, taken inclusively, does not exceed more than in the proportion of 3 to 2 the likelihood or chance of dying in a given year or half-year during the aforesaid vigorous period of life: so that taking, for example, two persons of equal constitution, one aged 40 years, and the other 58 years, if these two persons made such a contract that in case the person of 58 years shall happen to die in less than six months, the one aged 40 were to inherit a sum of 2000 florins from the property of the defunct; but that if on the other hand the person aged 40 years should die in less than six months, the one aged 58 were to have 3000 florins from the property of the deceased; such a contract cannot be considered disadvantageous for the person who would have the 3000 florins, if the event were favourable to him, and who in the contrary event would only lose 2000 florins.

I then presuppose that the greatest likelihood of dying in a given year or half-year of the second series of the ten following years (that is from 63 years to 73, taken one with the other, rather than in a given year or half-year of the period of the vigour of life), cannot be estimated at more than double, or as 2 is to 1; and as the triple, or as 3 is to 1, during the seven following years, that is from 73 years to 80. Finally, in supposing that life necessarily ends at the twenty-seventh year after the expiration of 50 years of age above presupposed, this time is neither assumed at too high nor too low a standard, as experience manifestly teaches us that the life of some men exceeds by a considerable period the age of 80 years, the age of 81 years and even more.

These three articles being presupposed, we have, by a demonstrative calculation, mathematically discovered and proved that the redeemable annuity being fixed at 25 years' purchase as above, the life annuity should be sold at 16 years' purchase, and even higher, to be in equality one with the other; so that in the purchase of one florin of life annuity, on a young and vigorous nominee, more than 16 florins should be paid, as is proved by the following demonstration.

We do not propose to follow the learned author through his demonstrations. We have accompanied him thus far from a belief that many even popular readers will desire to see something more than a mere outline of such an historically important document; and for the further purpose of showing how early the then young theory of mathematical prob. began to be applied to demonstrate the science of life contingencies. Nor is it necessary that we should follow De Wit's illustrations in detail, for Mr. Hendriks has prepared a summary of the result, which is at once concise, and comprehensive of the author's meaning:

Firstly. Out of 128 lives, aged say 3 years, 1 is supposed to die in every half-year of the first 100 halfyears or 2 per annum for 50 years, leaving 28 alive, aged 53 years, at the end of the term; out of whom 1 dies in every 9 months, being o'66 per half-year, during the next 20 half-years, or 133 per annum for 10 years, leaving 15:66 alive, aged 63 years, at end of second term; of whom I dies in every year for 10 years, being o'5 per half-year during the next 20 half-years, leaving 5'66 alive, aged 73 years, at the end of the third term; of whom 1 dies in every year and a half for 7 years, being o'33 per half-year during the next 14 half-years, leaving 1 alive, aged 80, at the end of the fourth term; which survivor does not live over another half-year. Secondly. Out of the 128 lives, those who die in the respective half-years between the ages of 3 and 80, will receive an annuity certain in half-yearly instalments, for a term equal in continuance to the number of completed half-years elapsed between age 3 and the date of their death; therefore, the sum of the present values of half-yearly annuities certain, for the corresponding terms multiplied into the numbers dying within such respective terms, gives the present worth of all the annuities which will be enjoyed by the 128 lives, 1/128 of which represents the present value of the single life annuity at age of, say, 3 years.

This system of valuation Mr. Hendriks finds to be identical with the fifth method described by Tetens.-(Assu. Mag., No. 1, pp. 9 and 18; No. 2, p. 18.)

We must make one or two further selections from the original document for the purpose of showing how some of the practical points incident to dealings in Life Contingencies were perceived from the very beginning. Thus, on the question of selection against the office: The person who for 16 florins has purchased 1 florin p.a. on a young, vigorous, and healthy life, has made a remarkably advantageous contract. I assert it to be remarkably advantageous for the following reasons: Because, in the first place, we have not been able to rate at a certain price, by perfect calculation or correct estimation, the power which the annuitant possesses (power which is of very great value to him), of choosing a life, or person in full health, and with a manifest likelihood of prolonged existence, upon whom to constitute or purchase his annuity; and there is much less risk or danger of a select, vigorous, and healthy life dying in the first half-year than in some of the following half years at the beginning of which the aforesaid life might perhaps prove to be in a weak state of health or even in a fatal illness; and such greater likelihood of prolongation of life in the purchase of an annuity upon a select, healthy, and robust life, may further extend itself to the second, third, and some other following it is, however, certain, when we examine the matter very scrupulously, that the likelihood of the decease of the nominees upon whom life annuities are usually purchased is less considerable and smaller in the first years after the purchase than in the subsequent years, seeing that the said life annuities are oftenest purchased and sunk upon lives of young and healthy children of 3, 4, 5, 6, 7, 8, 9, and 10 years or thereabout. During that time and for some years ensuing, these young lives, having become more robust, are less subject to mortality than about 50 years afterwards, and than for some years anterior to these 50 years; and so much the more as during the first aforesaid years they either are not, or are but little, exposed to external accidents, and extraordinary causes of death, such as those from war, dangerous voyages, debauch or excess of drink, of the sex, and other dangers; for females there are also confinements, and other like causes; so that the first years after the purchase or foundation of the annuity are the least dangerous, which is a considerable advantage for the annuitant, particularly if we reflect, as I have above stated, that one of the said first years may, as regards the original price of purchase, balance a great number of the subsequent years. Finally, and in the fourth place,

terms or half-years.

In the third place

so that the annuitant or his heirs were to receive 46 more entire half-years of annuity after the expiration of the term of the aforesaid 77 years, this could not, however, increase the price of the life annuity by more than 143 stuyvers of the same capital; and even if the annuitant could be assured that his heirs were, after the expiration of the above 100 years, to enjoy the life annuity from half-year to half-year, and that perpetually the value of the capital at the time of first purchase would not thereby be increased by ten stuyvers.

In consequence of all these reasons, we may assume it as established and demonstrated, that the value of a life annuity in proportion to the redeemable annuity at 25 years' purchase, is really not below, but certainly above, 16 years' purchase; so that a person wishing to purchase a life annuity in such proportion, and according to its real value, ought to pay more than 16 florins for 1 florin of annuity p.a.

of

Regarding the rate of int. employed in the calculation, he thus sagaciously observes: Besides the consideration that this calculation has been made on the principle of a deduction 4 p.c. p.a. at compound interest, and this with such benefit to the purchaser of the life annuity that he would realize not only the interest p.a., but also without any intermission interest upon interest at 4 p.c. p.a., as though he could always thus advantageously make use of his money in purchase of annuity; it is constant that one could not always find such opportunity of investing it, and that one is sometimes obliged to let it lie fallow for some time, and often to lend it at a materially smaller interest, to provide against a greater loss.

Finally:

Even besides this, as the capital of life annuities is not subject to taxation, nor to a reduction to a lower amount of annuity or interest, it follows, that if the blessing of the Almighty continue to be vouchsafed to this country, we may consider the life annuity as much more advantageous to the annuitant than the redeemable annuity. As may manifestly be judged by the example of foregoing times-by reflecting in fact that My Lords the States of Holland and West Friesland have in the course of a few years not only increased the charge for life annuities from 11 years' purchase to 12 years' purchase, and from 12 years' purchase to 14 years' purchase, but that these annuities have been sold, even in the present century, first at 6 years' purchase, then at 7 and at 8, and that the majority of all life annuities now current, and at the country's expense, were obtained at 9 years' purchase; which annuities by reason of the successive reductions of the rate of interest from 6 to 5 per cent., and then from 5 per cent. to 4 per cent., produce to the annuitants an actual profit of nearly one-half of such half-year's payment, and of more than one-half in the case of those annuities which were obtained at 8 years' purchase or under.

The Burgomaster of Holland and West Friesland in 1671, when the preceding report was made, was Herr J. Hudde, and he added a certificate to the report, which has caused some speculation: first, as to whether he understood the question sufficiently to impart any value to his certificate; and next, if he had any reservation as to some fault of a figure" which does not appear upon the surface: for there was a fault of a figure in the addition. Both points have more recently been decided, by competent authorities, in Hudde's favour.

Certificate: I the undersigned declare, that having attentively read and examined, at the request of my Lord the Grand Pensionary of Holland and West Friesland, the above propositions, and the conclusions thence derived, for finding the value of a life annuity compared with that of a redeemable annuity at 4 p.c. p.a. I am of opinion that the method employed for that effect is perfectly discovered, and that the conclusion made therefrom (namely that the purchaser of a life annuity still makes a gain in stipulating for 1 florin of annuity p.a. for 16 florins of capital) depends upon solid and incontestable mathematical foundations, provided that some fault of a figure has not been involuntarily made in the calculation of the Table, or in the copy, and addition of the items, which is an ordinary and well-known computation, but which I have not made.-J. Hudde.

The researches of Mr. Hendriks have estab. two facts of importance: first: that De Wit and Hudde were jointly prosecuting these inquiries and extending the calculations to two joint lives, and even three, four, five, or more lives; and next, that their attention was directed to the subject in consequence of the then recent discoveries in the doctrine of probability by Pascal and Fermat. We have, however, the authority of Mr. Milne for