-Whence u is r+ &c. Or u will be found (by a Process like the foregoing) = t 5n515 240 &c. the first Step of which, namely u=r+ or # for Practice. Loga Note, By how much the lefs the Logarithm of I will give you one Example of this Scholium. * Example.. If a 1s 1.06; the Question is what a is equal to? The Value of a may be with a great deal of Facility, in refpect of other Methods, found by Dr. Halley's Rational Theorem for evolving the Roots of Equations (of which I have treated in its proper Place, to wit in the Converging Series) by fuppofing g=1; for by proceeding by the faid Theorem with the faid Value of g, the Value of x will be found .0001596 and confequently the Value of g+x (g the 2d.) is = 1.0001596 which is pretty *See its Ufe in the following Part, Chap. 2. near équal equal to the Value of a; but if it be not near enough for your Purpose, renew the faid Theorem with the faid Value" of g the 2d 1.0001596, and the followingg (or g the 3d) will have the true Figures in it to a great many Places, that is it will be near equal to the Value of a but the Value of a may be more exactly had, or fooner, by fuppofing g 1, and by Dr. Halley's Irrational Theorem and its Corrections, of which I have treated in the Converging Series. But I will now fhew how to find the Value of a, expeditioufly and exactly enough, by the foregoing Scholium. Thus, α 1.06, by Hypothefis; therefore (by the Nature of Logarithms) 365 × L, a = L, 1.06 = .025305865264770240846731; wherefore .025305865264770240846731 L, a = 711699289991. 365 .00006933 137 Having thus found the Logarithm of a, a itself may be discover'd by the latter Part of our Scholium, by the help of the Table of Logarithms thus: Seek in the faid Table the nearest Logarithm without its Characteristick to the above Logarithm of a, and you'll find it to be 4.0000434; whose abfolute Number in the faid Table is 1.0001r: But the Logarithm of 1.0001 is .000043427276862669637313, which fubtracted from the above Logarithm of a, the Remainder is .000025903860849029652678; Therefore 000025903860849029652678 10000 &c. Indefinitely i =1: .00000000000001 76849054637 &c. I-.000059645843 &c. == 1.0001 +.00005965 35874706334020 &c. .0000000000000176859603⋅ &c. = 1.0001596535874529474417 &c. 'N. B. Part 16. begins Page 289. Signature P p. Intereft, or the Wife paid for the Loan of Money, may be either Simple, or Compound. Sed.. Of Simple Interek. Note,That a great part of this Part 16, has been Tranfcribed out of Mr. Ward's Young Mathematicians Guide, to Simple Intereft is that which is paid for which this Book is indebted for fome other There are feveral ways of computing (or answering Questions about) Simple Intereft; as by the fingle and double Rule of Three. Others make use of Tables compos'd at feveral Rates per Cent. As Sir Samuel Moreland, in his Doctrine of Intereft, both Simple and Compound, is all perform'd by Tables, wherein he hath [fays my Author] detected feveral material Errors committed by Dr. Newton, Mr. Kerfey upon Wingat, and Mr. Clavill, &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 raife fuch general Theorems as will fuit with all Cafes. In order to that, Let pany Principal or Sum pât to Interest r the Ratio of the Rate per 1 1. per Annum. t = the time of the Principal's continuance at Intereft: the amount of the Principal, and its Intereft. PP Note. Part XVI Note, The Ratio of the Rate is only the fimple Intereft of 11. for one Year, at any given Rate, and is thus found, Viz. 100 6:: 1 .. c,06 the Ratio at 6 per Cent. per Ann. Or 100.. 7::1 · 0,07 = the Ratio at 7 per Cent. &c. Again, 1007.5: 10,75 the Ratio at 7per Cent. &c. And if the given time be whole Years; then't the number of thofe Years: But if the time given be either pure Parts of a Year, or parts of a Year mixed with Years; thofe Parts must be turn'd to Decimals; and then - thofe Decimals, &c. Now the common parts of a Year may be eafily turn'd, or converted into Decimal parts, if it be confidered, These things being premifed, we may proceed to raising the Theorems. Letr the Intereft of 11. for one Year (as before) then 2r the Interest of 1 1. for two Years. 3r the Intereft of 11. for three Years. and 4r the Intereft of 11. for four Years: And fo on, for any number of Years propofed. Hence it is plain, that the Simple Intereft of one Pound is a Series of Terms in Arithmetick Progreffion Increasing, whofe firft Term and common Difference is r, and the number of all the Terms is ; therefore the laft Term will always be tr the Intereft of 11. for any given time fignified by Then As one Pound is to the Intereft of 17. 1: fo is any Principal or given Sum to its Intereft; That is, 1. tr :: pptr the Intereft of p: Then the Principal being added to its Intereft, their Sum will be A the amount required: Which gives this general Theorem. 1.Theorem trp + p =A=Pxtr+i. From whence the three following Theorems are easily de duced. |