Or new r=283, which being involved, &c. will appear to be the true Root, that is, a 283 juft. Note, Thefe are ufually called the three Forms of Cubick Equations; and in the Solution of the third or laft Form, viz. ba-aaa = G, you may meet with fome feeming Difficulties; especially in making Choice of the first r, because this Equation is an ambiguous Equation, and hath two Affirmative Roots, viz. a greater and leffer Root. But having once found either of them, the other may be eafily obtained by Divifion only; as in the Quadratick Equations. Vide Chap. 8. As for inftance, in the laft Example, a 283 and 123456 aa aa 12272861. Make these two Equationso, to wit, let a-2830, and ➡aaa+123456 a—122728610. Then, a-283) —aaa+123456 a—12272861 (-aa -aaa+283a a Hence it appears that -aa-283a+433670. Confequently aa+283a=43367 this Equation being folved, a=110, 2722 &c. which is the leffer Root of the aforefaid Equation ba —a a a — G, &c. After this Manner all the poffible and impoffible Roots of any Equation may be eafily difcovered, any one of it's Roots being once found. I fhall therefore omit inferting more Examples of that kind. Suppofe aaa+baa+ca G. Quære a. Let b=74, c=8729, and G=560783. By Trial (as before) it will be found that the next nearest r40 being something less than just. Therefore Therefore r+e=a I X C 2 in Numb. 2cr+ce=ca 3brr+2bre+bee=baa 3 in Numb. 6118400+ 5920e + 74ee 1011e = Operation 100,2 += I Ift Divifor 101,2) D 1,01 Or new r= 41,5 for a fecond Operation, which being duly involved, &c. will be found more than juft. Therefore 1|r-e=a Then { 2crce=ca 3 brr-2bre+bee=baa These being turned into Numbers, &c. as above, they will be 20037.75€ 198,5 e e 390,375, which being divided by 198,5 the Co-efficient of, will become 100,946e-ee= 1,966624, &c. = D. Operation 100,946 41,5000000=r 1,00936 41,4805153=re = a 957264 908343 489210 403708 855020 807416 Let the last Equation in the Enigma, Chap. 9. be here proposed for a Solution. Viz. aa aa + ba aa—ca a➡da⇒G; b=2, c=288, d= 506, and G 1513, Quære a. By Tryals it will be found, that the next nearest r=20, being fomething more than just. I3 xb 4brrr - 3 brre + 3breebaa a These being turned into Numbers, and thofe duly collected, according as the Signs of the Equation direct, they will become 50680- 22374e2232ee1513, which being all divided by 2232 the Co-efficient of ee, will be 10 eee 22 — D, By what hath been already done about the Solution of these few Equations (being carefully obferved) I prefume the Learner will easily conceive how to proceed in the Solution of all Kinds of Equations, be they never fo high, or adfected; therefore I fhall not here propofe many various Examples, but only take them as they fall in Courfe, when I come to the next Part, wherein you will (perhaps) find fuch Equations with their Solutions as are not common. CHAP. IN CHAP. XI. Of Simple Intereft, Annuities, or Penfions, &c. NTEREST, or the Ufe paid for the Loan of Money, is either Simple; or Compound. Sect. 1. Of Simple Interef. IMPLE Intereft, is that which is paid for the Loan of any Principal or Sum of Money, lent out for fome Time, at any Rate per Cent: agreed on between the Borrower and the Lender; which, according the late Laws of England, ought to be fix Pounds for the Ufe of 100% for one Year, and twelve Pounds for the Ufe of 100l. for two Years: and fo on for a greater, or leffer Sum, proportionable to the Time propofed. There are feveral Ways of computing (or answering Questions about) Simple Intereft; as by the fingle and double Rule of Three (See Page 96, &c.) others make Ufe of Tables composed at several Rates per Cent, as Sir Samuel Moreland, in his Doctrine of Intereft, both fimple and compound, all performed by Tables; wherein he hath detected feveral material Errors committed by Sir Ifaac Newton, Mr Kersey upon Wingate, and Mr Clavil, &c. in the Bufinefs of computing Intereft, &c. by their Tables, too tedious to be here repeated. But I fhall in this Tract take other Methods, and fhew that all Computations relating to Simple Intereft are grounded upon Arithmetick Progreffion; and from thence raise fuch general Theorems, as will fuit with all Cases. In order to that Let Pany Principal or Sum put to Intereft. Rthe Ratio of the Rate, per Cent. per Annum. t = the Time of the Principal's Continuance at Intereft. A the Amount of the Principal, and it's Interest. Note, The Ratio of the Rate, is only the Simple Interest of 1. for one Year, at any given Rate; and is thus found. Viz. 100: 6 :: 1:0,06 Or the Ratio at 6 per Cent. per Annum. the Ratio at 7 per Cent. &c. the Ratio at 7 and per Cent. And if the given Time be whole Years; then the Number of whole Years: but if the Time be given, be either pure Parts of a Year, or Parts of a Year mixed with Years; thofe Parts must be turned into Decimals; and then t thofe Decimals, &c. Now Now the common Parts of a Year may be easily turned or converted into Decimal Parts, if it be confidered Thefe Things being premifed, we may proceed to raising the Theorems. Let R Then 2 R the Intereft of 11. for one Year, as before. the Intereft of 1 1. for three Years. = for any Number of Years propofed. And fo on Hence it is plain, that the Simple Intereft of one Pound is a Series of Terms in Arithmetic Progreffion increafing; whofe firft Term and common Difference is R, and the Number of all the Terms is t. Therefore the last Term will always be t R = the Intereft of 1 1. for any given Term fignified by t. As one Pound is to the Intereft of 11. fo is any Then {Principal or given Sum: to it's Intereft. Then That is, 1.:tR:: PtRP the Intereft of P. the Principal being added to it's Intereft, their Sum will be = A the Amount required: which gives this general Theorem. Theorem 1. t RP+P = A From whence the three following Theorems are eafily deduced. A-P - =P. Theorem 3. =R. A-P Theorem 4. RP Thefe four Theorems refolve all Queftions about Simple Intereft, Question 1. What will 2561. 10s. amount to in 3 Years, one Quarter, 2 Months, and 18 Days, at 6 per Cent. per Annum. Here is given P = 256,5; R = 0,06; and t = 3,46599 Quære A. per Theorem 1. For 3 Years one Quarter 3 2 Months 0,16667 0,08333 × 2 18 Days 0,04932 0,00274 x 18 Hence 3,46599: × 0,06 0,2079594 =t R Then 0,2079594 x 256,553,341586t RP = And 53.341586 +256,5 309,841586 t RP + P = A. 309,841586tRP+P That is, 309,841586 = 309 /. 16 s. 10 d. being the Answer required. Question |