1.50363) 37.7723 ( 75=u; that is u≈ 75. the Yearly Rent required by the Question. Thefe two Examples of finding p and u, do fully fhew the Method that must be used in Refolving the two General, and indeed the most Useful Questions about Annuities or Leafes in Reversion= And if there be occafion, either the Rate or the Time; viz. R or t, may be found by a due application of their respective The orems. Note, If the Rents or Annuities, &c. are to be paid Halfyearly, or Quarterly, that then R=1.06) =1.06 and R=1.06 for Half-yearly Payments, at for Quarterly, &c.6 per Cent. Thus far concerning fuch Annuities or Leafes, &c. that are limited by any affigned Time; and 'tis only fuch that can be comfuted by Theorems or certain Rules. However, it may not per"haps, be unacceptable to Infert a Brief Account of fome Estimates that have been reafonably made by two Ingenious Perfons, about the Proportion or Difference of Men's Lives, according to their Several Ages, which may be of good Ufe in computing the Values of Annuities, or taking Leafes for Lives, &c. Sir William Petty in his Difcourfe made before the Royal Society (Anno 1674.) concerning the ufe of Duplicat Proportion, in the Life of Man and its Duration; faith, That it is found by Experience, there are more Perfons Living of between Sixteen and Twenty-fix Years Old, than of any other Age or Decad of Years in the whole Life of Man (viz. 70 or 80 Years), His Reafons for that Affertion I fhall omit: But fuppofing it True, He thence infers, that the Roots of every Number of Men's Ages under Sixteen (whofe Root is 4) compared with the faid Number A, doth fhew the Proportion of the likelihood of such Men's reaching to the Age of 70 Years. As for Example; 'Tis 4 times more likely, that one of 16 Years Old fhould live to 70, than a New-born Babe; 'tis 3 times more likely, that one of 9 Years Old fhould attain the Age of 70, the faid Infant, &c. than Qn Part XVI. On the other Hand, 'tis 5 to 4, that one of Twenty-five Years Old will Die before one of Sixteen: And 6 to 5, that one of Thirty-fix will die before one of Twenty-five: And fo on, according to the Roots of any other declining Age, compared with (4, 58, &c.) the Root of 21, which is the Year of Perfection according to the Senfe of our Law, and the Age for whofe Life a Leafe is moft Valuable. 2. The Ingenious and great Mathematician, Captain Edmund Halley, (in Philofoph. Tranfact. Numb. 196.) doth with great Industry and Skill, draw an Estimate of the Proportion of Men's Lives from the Monthly Tables of the Births and Funerals in Breflaw, the Capital City of the Province of Silefia: Whence He proves, that it's 80 to 1, a Person of Twenty-five Years Old will not Die in a Year: That 'tis 5 to 1, That a Man of Forty will live Seven Years: That a Man of Thirty Years Old may reaJonably expect to live Twenty-Seven, or 28 Years, &c. Now, from thefe and the like Proportions (he justly infers that) the Price of Infurance upon Lives ought to be regulated, there being a great Difference between the Life of a Man of Twenty and one of Fifty; for Example, 'Tis 100 to 1, that a Man of Twenty dies not in a Year, and but 38 to 1, for a Man of Fifty Years of Age. And upon thefe alfo depends the Valuation of Annuities for Lives: For it is plain, that the Purchaser ought to pay only fuch a part of the Value of any Annuity, as he hath Chances that he is Living. And for that purpose, he hath taken the pains (which was not a little) to compute the following Table, that fews the Value of Annuities for every 5th Year of Age to the 70th. Age. Year's Purchafe. Age. Year's Purchafe. Age. Year's Purch. Sect. 4. Of Purchafing Freehold or Real Eftates, at All Freehold or Real Eftates, are fuppofed to be Purchased or Bought to continue for Ever, (viz. without any limited Time) therefore the Business of computing the true Value of fuch Eftates,is grounded upon a Rank or Series of Geometrical Proportionals continually Decreafing ad Infinitum. Thus, let p,u, R,denote the fame Data as in the laft Section ; Example. Suppofe a Freehold Eftate of 751. Yearly Rent were to be Sold; What is it Worth, allowing the Buyer 6 per Cent, &c. Compound Intereft for his Money? In this Queftion there is given u 75, R1.c6 to find p per Theorem 1. Thus ; R−1=.06) 75=4(1250l. =p the Answer required. And fo for any of the reft, as occafion requires: But if the Rent is to be paid, either by Half-yearly or Quarterly Payments, then R=√1.56 for Halfyearly And RVV1.06 for Quarterly Sf Half-2 Payments at 6 pér S Cent. Or { > Payments at 8 per S Cent. R1.08 for Yearly RV1.08 for Half-yearly RV:V1.08 for Quarterly R= The Like is to be understood for any other propofed Rate of Intereft either Greater or Less than 6 per Cent.. The Application of thefe Theorems to Practife is fo very Eafj, that it's needless to Infert more Examples. PART XVII. Some of Diophantus's Queftions. Quest. 1. To find two Square Numbers Equal to a given Square Number=dd. * 1. Take any two Unequal Numbers; Suppofe s the Greater, and r the Leffer; then for the fide of the first Square fought, put ra. 2. And for the fide of the fecond Square fought put sa — d or d-sa. 3. Then from the first Step, the firft Square is r2 a2. 4. And from the fecond Step, the fecond Square is 2 d s a + s saa. 5. Therefore the Sum of the Squares fought is s2 a2 - 2 dsa + dd. dd r2 a2+ dd. 6. Which Sum must be Equal to the given Square Hence this Equation arifeth ra2 + s2 a2 — 2 d s a + dd. d d. - 7. Which Equation after due Reduction, gives a = 2 ds sst rr 8. Therefore, by the firft and feventh Steps, the fide of the firft Square fought is now made known, for it is = 2rsd sstir 9. And by the fecond and seventh Steps the fide of the fecond Square is also made known, for it is ssd-rrd {strr Which two laft Steps gives this. For if swas r then the fecond Square would be,and the firt would be the gi ven one. |
