And, by Dr. Halley's Rational, or Irrational Theorem, x .0002126; therefore g+x=. 9362126=y nearly; Confequently R=1.068133+. And then it will be 1..1.068133-1:: 100..6.8133 the Rate per Cent. &c. as was required. Thefe four Questions include all the Variety that can be propos'd about purchafing Annuities, or Leafes, &c. which are either immediately to be enter'd upon, or in Poffeffion at the Time when the Purchase is made. But Theorems for folving fuch Queftions as relate to Annuities, or taking Leafes, &c. in Reverfion may be raised in the following Manner. 1. Suppofe it was required to compute the prefent Worth of 751. Yearly Rent which is not to commence or be enter'd upon until 10 Years hence, and then to continue 7 Years after that Time, at 6 per cent. &c. compound Interest ? Here 75, t=7, R=1.06; and fuppofe T-10 the Number of Years the Annuity is not to commence, or be enter'd upon: Then By the 1st Theorem of this 3. the prefent Worth of u Pounds Yearly Rent to continue T--t Years at R-1 per 1. And by the faid Theorem, the prefent Worth of u Pounds Yearly Rent to continue T Years at the fame Rate of Inte R-I *This in Con- * Confeq. the prefent Worth required, viz. clufion, is the Jame with Mr. Ward'sMethod. 264. See Page น Theorem.I. Where By this Equation the preceeding Queftion may be folv'd; thus LR txLR= 7 x L 1.067x.0253059.1771410; I "L===L1—tLR= 1.8228590 And the Number answering to this Log. is .6650572; TXLR 10x.02530586.2530586 L:R-1:-L.06 2.7781512 Sum 1.0312098 Lp: Confequently p233.7882331. 15 s. 9 d. the prefent Worth of 751. per Annum in Reverfion, &c. as was required. Ex. 2. What Annuity or Yearly Rent to be enter'd upon 10 Years hence, and then to continue 7 Years may be purchafed for 233. 15 s. 9 d. ready Money at 6 per Cent. "&c. compound Intereft? The Theorem for folving this Question deducible from RTX: R-1: the foregoing Theorem 1. ist 1 Rt Confeq. Lp+TxLR+L:R~1:~L:1 Lp L 233.788-2.3688221 TXLR .2530586 = L:R-1: 2.7781512 Sum 1.4000319 u. Theor. 2. Lu 1.8750613u75; that is, 751. is the Yearly Rent required by the Queftion. *B b 2 These Thefe two Examples of finding p and u do fully fhew the Method to be ufed in refolving the two general, and indeed moft ufeful Questions about Annuitics or Leafes in Reverfien: And, if there be Occafion, either the Rate or the Time, viz. R, t or T may be found by a due Application of their refpective Theorems. Note, If the Rents or Annuities, &c. are to be paid Half Yearly, or Quarterly, that ་ Then R1.c6 for Half-Yeatly > Payments at 6 per And R1.06 for Quarterly S Cent. &c. &c. Sect. 4. Of purchafing Freehold or real Eftates at Compound Interest. All Freehold or Real Eftates are fuppos'd to be purchas'd or bought to continue for ever (viz. without any limited Time); therefore the Bufinefs of computing the true Value of fuch Eftates is grounded upon a Rank or Series of Geometrical Proportionals continually decreafing ad infinitum : Thus let, u, R denote the fame Data as in the last Section; ut fupra. Theor. 2. pR-pu Example, fuppofe a Freehold Eftate of 751. Yearly Rent were to be fold; what is it worth, allowing the Buyer 6per Cent. &c. compound Intereft for his Money? In this Question there is given And p per Theorem 1. Thus = 75, R= 1.06; to = R-1.06) 75" (1250l. p the Anfwer 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; Cent. &c. Then R 1.06 for Half-yearly Payments at 6 per Payments at 8 per The like is to be understood for any other propos'd Rate of Intereft either greater or lefs than 6 per Cent: &c. The Application of thefe Theorems to Practice is fo very eafy, that it is needlefs to infert any more Examples. PART IT PART XVII. Some of Diophantus's Questions. T may be expected that 'tis poffible to folve the following Queftion if propos'd in more univerfal Terms; viz. that 'tis poffible to divide any given Number into two Rational Squares: But that it is not Mr. Thomas Wallis of the City of Corke has demonftrated; thus Lemma.1. All even fquare Numbers are divifible [viz. divifible without leaving any Remainder] by 4. Demonftration. The Roots of all even Squares are even; therefore divifible by 2. the Let* 2 n be NB. All the Root of any even Square. The Square of 212 Symbols or Let- is 4nn; wherefore all even Squares are divima's, and in the fible by 4. ters in thefeLem following Con- Lemma 2. Any odd Square Number dividfeet, and Scho- ed by 4 leaves a Remainder of 1. bers. I lium are fup- Demonftration. The Roots of all odd pofed to be equal Square Numbers are odd. Let 2 n+1 be= to integer Num- the Root of any odd Square. The Square of 2n+1=4nn+4n+1; therefore any odd Square divided by 4 leaves 1. [for 4nn--4-|-1 divided by 4 gives nn-22 for the Quotient, and 1 for the Remainder.] Lemma 3. If a Number confift of two even Squares it will be divifible by 4. This is evident from Lemma 1. Lemma 4. If a Number confifting of two odd Squares be divided by 4 it will leave a Remainder of 2, by the 2d Longma. Lemma 5. If a Number confifting of an even and an odd Square be divided by 4 it will leave a Remainder of 1. This Lemma is evident from the 1st and 2 d Con |