^{page 224 note *} In the History of Life and Death, it may be perceived that Lord Bacon has not given the most obscure hint of an attempt to discover an approximation to the laws of vitality or mortality, such as they are now understood. H e restricts himself entirely to a curious collection of instances of human longevity, and to dissertations (after the fashion of the old thaumaturgists) respecting the prolongation of life by their usual specifics, some of which are common-place and indubitable, whilst the more extraordinary are replete with that degree of absurdity, that it is a marvel how they could have obtained further credit in the progress of the Elizabethan age.

^{page 226 note *} Vide “A Discourse upon Usurie, by waie of dialogue and oracions, for the better varietie and more delight of all those that shall read this treatise: by Thomas Wilson, doctor of the Civil lawes, one of the maisters of his maiesties honourable courte of requests. Imprinted at London by Roger Warde, dwelling neere Holburne Conduit, at the signe of the Talbot.” 1584. (12mo. litt. goth.) Fol. 104. 6.

This work was composed at least 15 years previously; the Epistle dedicatory to “his most especiall and singuler deere Lord, the Earle of Leicester, &c. &c.” being dated “From the Queenes maiesties Hospitall at Saint Katreynes, thys twenty of July, 1569.” The author (Sir Thomas Wilson) was Tutor to Henry and Charles Brandon, sons of the Duke of Suffolk, and to the Princess Mary; Ambassador from Queen Elizabeth to Mary Queen of Scots, Secretary of State, Privy Councillor, and Dean of Durham. He died in 1581. (See Bibliotheca Grenvilliana.) In another work it is stated, that “in the reign of Mary he lived abroad, and was seized by the Inquisition at Borne, but escaped in consequence of a fire, which induced the populace to force open the dungeon that the prisoners might not be burnt.” I am sure that the Holy Brotherhood could not fairly have taken him to task, if it had but known how zealous and orthodox a defender of its own views. on the subject of interest of money existed in the person of Wilson, who took quite the Romanist and Canonical side of the question, and in his treatise does not so much as refer to the amended and faithful teaching on the subject by the great promoters of the Reformation. The discourse is one of great prolixity, occupying upwards of four hundred pages, in a Dialogue entitled “A Communication or Speeche betweene the riche worldlie merchant, the godlie and zealous preacher, the temporall and civil Lawiers, touching usurie, or the lone of monie for gaine.” It will well repay perusal, and its earnestness is worthy of attention. It forms moreover a tolerably complete manual of the extraordinary quibbles and make-shifts by which the money trade of the 16th and preceding centuries was conducted; and all this is seasoned with a running artillery of theological denunciation, supposed allusions of Holy Writ, and direct attacks and anathemas of popes and councils.

^{page 229 note *} Cleirac's comment on another prohibitory clause, that against insuring perishable property, runs thus:—Nee mors humano subjacet arbitrio. Contre les effets du temps, de l'âge, et de la nature, toutes les asseurances et précautions humaines sont vaines et fort inutiles.”

^{page 230 note *} The business of Marine Insurance also met with opponents in France in the last century. The anonymous author of “Les Intérêts de la France mal entendus” (a work which attracted much attention, and in other countries, and which was highly lauded for its patriotic views) observes, “Quoi qu'il en soit, je dirai hardiment que les assurances chez nous ont mis des entraves aux progrès de notre Commerce, et que si notre Administration n'y remédie, elles sont à la veille de causer son entière ruine.” He goes on to argue that the world had done without assurances for 6000 years, and therefore might just as well dispense with them in his time. (!!) (See pages 46 to 59, Edition of 1757.) The reputed author was, I believe, the Chevalier Ange Goudard, of Montpelier.

^{page 232 note *} The title, in the original, is here “Waardye van Lyf-renten naer proportie van Losrenten.” Baily and other English writers, who have called it “De Vardye Tan de Lif-renten,” &c., seem to have followed the French misquotation.

^{page 238 note *} In the original, the old galley form of Division (or, as the Dean of Ely terms it, the scratch method of division) is here employed, and also in the subsequent examples. I have not, however, thought it worth while to trouble the printers with the now strange and obsolete type it requires.

^{page 243 note *} The above Table, computed to such a nicety by De Wit's directions, is composed of the progressive summations of the present values of 1 Million Florins or 20 Million Stuy-vers per annum, receivable in 100 half-yearly instalments for 50 years. The second and every even term will be found correct, on the supposition of discount at 4 per cent, per annum; but the first and every odd term erroneous, in the same way that the remark is applicable to Smart's and Tetens' (or Von Drateln's) Tables, at intermediate half-years, by reason of the interest being reckoned by a geometric instead of by an arithmetic mean. —In the original a complete table is given from 1 to 200 half-years, which, however, it s i useless to repeat in full, as the even terms may be obtained by an easy process from the data in other works, and the odd terms are inapplicable to modern purposes.

Struyck, in his Uitrekening van de Lyfrenten, has some remarks on the “prodigious labour” of the two Bookkeepers who calculated the Table, although when we compare it with similar ordinary computations of more modern times it is relatively not worthy of such an appellation. At the present date, the tendency is certainly to underestimate such labours; a reaction to the juste milieu may, however, take place after a surfeit of Statistics.

^{page 244 note *} The addition here presents a clerical error. It should give 281, &c, instead of, as above, 280, &c.; and the general summation 409, &c., instead of 408, &c. This proceeds no further; for in the working of the annuity valuation, th e total is correctly given by De Wit.

I particularly refer the reader to a further note on the subject, at the end of this treatise of De Wit; where I have thought it right to make a few observations on the Burgomaster Hudde, and his certificate, which hears in rather a singular manner on the above casual mistake in the addition.

^{page 244 note *} 40,964,113,736 is here correctly given by De Wit.

^{page 246 note *} De Wit's calculation may be simplified and explained as follows:—Firstly. Out of 128 lives, aged say 3 years, 1 is supposed to diein every half-year of the first 100 half-years, 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 0·66 per half-year during the next 20 half-years, or 1·33 per annum for 10 years, leaving 1466 alive aged 63 years at endof the second term; of whom 1 dies in every year for 10 years, being 0·5 per half-year during the next 20 half-years, leaving 1466 alive aged 73 years at end of the third term; of whom 1 dies in every year-and-a-half for 7 years, being 0·33 perhalf-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 thenumber of completed half-years elapsed between age 3and 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 orall the annuities which will be enjoyed by the 128 lives, 11/28 of which represents the present value of the single-life annuity at ageof, say,3 years. The system of valuation is therefore identical with the fifth method described by Tetens, whose formula I have hadthepleasure to refer to on a previous occasion. (See the Assurance Magazine, No.I., p. 9, and 18; and No. II., p. 18.)

If arranged in themodern form of a life table, the following abstract would represent the course of the results of De Wit's suppositions as to mortality:—

^{page 250 note *} It is difficult to attempt to fathom the inducements for the latter clause of limitation in the Burgomaster Hudde's certificate. The clerical error in the mere addition, and not the application or final result of De Wit's table for the valuation of an annuity, seems t o be what is there referred to by Hudde. We have briefly mentioned this lapse of the pen, in a former note; and it is here necessary to state, that not only is it shown to be no more than that, by the present test of a fresh addition of the items, hut the accuracy of the applied or amended final total is amply proved by the application of two formulæ for the summation of temporary annuities increasing in the order of the natural numbers. The motive for such a trial has been simply the apprehension that the terms in which Hudde's certificate is couched might, in the absence of explanation, throw a primâ facie doubt on his concurrence in the absolute result, though not on the principle of De Wit s calculation; or, otherwise, it might lead to the idea that it was an invidious remark on the part of Hudde, who could have tested the result. Of De Wit's accurate application of the data laid down, the examination of the figures leaves no question; and respecting the terms employed by Hudde, we ought to be willing to suppose an explanation, in the likelihood of his having been requested by the Grand Pensionary to give his certificate on this new and very remarkable computation, at the last moment, before its presentation to the States-General; thus not leaving him the needful time to examine it in all its details. This view becomes the more reasonable, when we discover what is said of the busy responsibilities of Hudde's municipal duties, the anxieties of which, in those trying times of the history of the Netherlands, must have been of pressing importance. As confirming this, I extract a passage which occurs in a recent German Biography of Leibnitz:—“Ich habe, in. Amsterdam, mit Hudden gesprochen, welchem aber die Geschäfte fur den Staat alle Zeit wegnehmen. Denn er ist einer von den zwolf Bürgermeistern der Stadt, welche nach einander die Regierung führen; kurzlich war er regierender Büraermeister, jetzt ver-waltet er das Amt ernes Schatzmeisters. Es ist gewiss, dass seine Papiere noch sehr vortreffliches aufbewahren. Die von Slusius bekannt gemachte Methode der Tangenten war ihm lange bekanntgewesen,” &c. (Extract from letter of Leibnitz to Oldenburgh, dated 18/28 November, 1676, and quoted in a letter from Collins to Sir Isaac Newton.) See Guhra-uer's Gottfried With. Freiherrvon Leibnitz. Breslau, 1846. 1 vol. p. 183.Google Scholar

^{page 250 note †} I here quote the original passage, apprehending that it would lose much of its epigrammatic terseness in any translation:—“J'ai presque ri des airs que M. le Chevalier de Méré s'est donné dans sa lettre à M. Pascal, que M. Bayle rapporte au mêne article.

Mais je vois que le Chevalier savoit, que ce grand Géie avoit ses inégalités, qui le ren-doient quelquefois trop susceptible aux impressions des spiritualistes outrés, et le dégou-toient mêne par intervalles des connaissances solides, ce qu'on a vu arriver depuis, mais sans retour, à Messieurs Stennonis et Swammerdam, faut e d'avoir joint la métaphysique véritable à la physique et aux mathématiques. M. de Méré en profitoit pour parler de haut en has à M. Pascal. Il semble qu'il se moque un peu, comme font les gens du raonde, qui ont beaucoup d'esprit et un savoir médiocre. Ils voudroient nous persuader que ce qu'ils n'entendent pas assez est peu de chose. Il auroit fallu l'envoyer à I'école, chez M. Roberval. Il est vrai cependant que le Chevalier avoit quelque génie extraordi-naire, même pour les mathématiques, et j'ai appris de M. des Billettes, ami de M. Pascal, excellent dans les Méhaniques, ce que c'est cette découverte, dont ce Chevalier se vante ici dans sa lettre, C'est qu'étant grand joueur, il donna les premières ouvertures sur 1'es-time des paris; ce qui fit naître les belles pensées De Alea, de Messieurs Fermat, Pascal, et Huygens, oú M. Roberval ne pouvoit, ou ne vouloit rien comprendre. M. le Pension-naire de Wit a poussé cela encore davantage, et l'applique à d'autres usages plus considera-bles, par rapport aux rentes de vie; et M. Huygens m'a dit, que M. Hudde a encore eu d'excellentes méditations là-dessus, et quec'est dommage quil les ait supprimées avec tant d'autres. Ainsi les jeux mêmes mériteroient d'être examinés; et si quelque mathéinati-cien pénétrant meditoit la-dessus, il y trouveroit beaucoap d'importantes considérations; car les hommes n'ont jamais montré plus d'esprit que lorsqu'ils ont badinés.”

^{page 254 note *} Inleiding tot de Algemeene Geographic, benevens eenige Sterrekundige en andere Verhandelingen. Door Nicolaus Struyck. 4to. Amsterdam, 1740. The author was a physical geographer and astronomer, and Fellow of the Royal Society of London. His work embraces a variety of subjects; and about 50 pages at its conclusion are devoted to that of life annuities. It has, therefore, bee n misquoted in those instances where we see this section mentioned as comprised in Struyck's Introduction to his Universal Geography. In the year 1753 he published a continuation of his researches in political arithmetic, under the title of Nader Ontdekkingen noopens den Staat van het Menschelyk Geslagt, in his work Vervolg van de Besehryving der Staartsterren, &c. W. Kersseboom, of the Hague, published some strictures on the first work, in 1740. (Eenige Aanmerkingen op de Gissingen over den staat van het menschelyk Geslagt, Uitreekening van de Lyfrenten, &c. 'S Graven-hage, 4to., pp. 18.)