Rule. Take fo many of the first Figures of the proposed Number as your Table of Logarithms extends to; and to the Number exprefs'd by thefe Figures add 1: Then place fo many Cyphers after this Sum, as alfo the fame Number of Cyphers after the faid firft Figures, that thefe Numbers may be one of them greater, and the other lefs than the propos'd Number. Now the Logarithms of the greater and less Numbers may be found by the Table, and from the Confideration that the Logarithms of a x 10, a x 100 are equal to La+1, La+2 refpectively: Wherefore fay As the Difference of the greater and lefs Numbers is to the Difference of their Logarithms, fo is the Difference of the propos'd and lefs Numbers to the Difference of their Logarithms, which added to the Logarithm of the lefs Number gives the Sum the Logarithm fought of the propos'd Number nearly. Examples. 1. Let it be required to find the Logarithm of 123459 by the Help of a Table of Logarithms from 1 to 100000. The first Figures of this Number which this Table extends to are 12345, to which I being added, the Sum is 12346: And Ι the propos'd Number is 123459; confeq. The greater Number is 123460, whofe Log. is 5.0915263 L12346-|- 1 And the lefs Number is 123450 whofe Log. is 5.0914911 =L12345+ı Wheref. as 10.. ΙΟ .0000352 .0000352::9. .00003168.0000317The Logarithm of 123459 is 5.0915228 Anf. 2. Let it be required to find the Logarithm of £234598 by the help of the abovementioned Table. The first Figures of this Number which this Table extends to are 12345, to which I being added the Sum is 12346: And The The propos'd Number is 1234598; confeq. The greater Number is 1234600 whofe Log. is 6.0915263 =L12346+2 And the lefs Number is 1234500 whofe Log. is 6.0914911 L12345+2 100 98 .0000352 Wheref. as 100.. .0000352::98.. .000034496.0000345The Logarithm of 1234598 is 6.0915256 Answer. Note, The Logarithm fine of any Number of Degrees, Minutes, Seconds, and, in fome Cafes, Thirds may be found by the help of the above Rule, and of the common Table of Logarithm - Sines, &c. instead of that of the Logarithms; thus Suppose it was required to find the Logarithm fine of 38 40 55 51 The firft Figures of this Number, which this Table ex tends to, are 38 40, to which I being added, the Sum is 38 41: And " The propos'd Number is 38 40 55 51; confeq. The greater Number is 38 41, whofe LS is 9.7958909 And the lefs Number is 38 40, whofe LS is 9.7957330 Now fay as I.. ,0001579::55 51 .. The Logar. - fine of 38 40 55 51 is .0001470 9.7958800 Nearly Answer. 2. To find the Number that answers to any given Logarithm to one or two Places of Figures more than your Table extends to, by the Help of a Table of Logarithms from 1 to at leaft 10000. Rule. When the Logarithm given cannot be exactly found by your Table, take the two nearest Logarithms that are greater and lefs, as alfo the Numbers answering those Logarithms, from your Table: Then fay, As As the Difference of the nearest greater and lefs Logarithms is to the Difference of the Numbers anfwering these Logarithms, fo is the Difference of the given and lefs Logarithms to the Difference of the Numbers anfwering these Logarithms, which, added to the Number anfwering to the lefs Logarithm, gives the Number required nearly. Examples. 1. Let it be required to find the Number anfwering the Logarithm. 4669347 by a Table of Logarithms from 1 to 100000. The given Logarithm is The nearest that is greater is the Log. of 2.9305.4669417 And the nearest lefs is the Log. of .4669347 2.9304.4669269 As .0000148.. .0001 :: .0000078.. .0000532.930453-Anfwer. 2. Let it be required to find the Number of Degrees, Minutes, Sec. and Thirds answering the Log. - fine 9.7958800 The nearest greater is the Log. fine of 38 41 9.7958909 9.7957330 -=000 55 51 As .0001579.. 1:: .0001470.. .931- Nearly Answer. PART XVI; PART XVI. Of INTEREST. Intereft or the Ufe paid for the Loan of Money may be either Simple or Compound. Chap. I. Of Simple INTEREST. 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 formerly according to the Laws of England, was 6 Pounds for the Ufe of 100 Pounds for one Year, and 12 Pound for the Ufe of 100 Pound for two Years: And fo on for a greater or leffer Sum proportionable to the time propos'd. There are feveral Ways of computing (or anfwering Queftions about) fimple Intereft; as by the fingle and double Rule of Three. Others make ufe of Tables compofed at feveral Rates per Cent. But I fhall in this Tract fhew that all Computations relating to fimple Intereft are grounded upon Arithmetical Progreffion; and from thence raise fuch general Theorems as will fuit with all Cafes. In order to that Section 1. pany Principal or Sum put to Interest. Let r the Ratio of the Rate per Cent. per Annum. t = the Time of the Principals Continuance at Interest a the Amount of the Principal and its Interest. Note, the Ratio of the Rate is only the Simple Interest of 1 Pound for one Year at any given Rate, and is thus found Viz. 100.. 6:: I 0.06 the Ratio of 6 per C. p. An. 0.07 the Ratio of 7 per Cent, &c. Again100..7.5 I.. 0.075 the Ratio of 7 per Cent, &c. Or 100 .. 7 ፡፡ 1 * Z And 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 mix'd with Years, thofe Parts must be turn'd'into Decimals; and then t thofe Decimals, &e. Now the common Parts of a Year may be eafily turn'd or converted into Decimal Parts, if it be confider'd that one. Day is the 35 Part of a Year = .00274 fere Month is the Part of a Year.0833333 &c. Quarter is the Part of a Year.25 Half a Year. 5. And three Quarters.75 + Thefe Things being premised, we may proceed to raising the Theorems. Let the Intereft of 1 Pound for one Year as before, Then 27 the Intereft of 1 Pound for two Years. And 3r the Intereft of 1 Pound for three Years, 4r= 4 the Intereft of 1 Pound for four Years. And fo on for any Number of Years propos'd. Hence it is plain that the fimple Intereft of 1 Pound for any Time, or number of Years fignified by t istr Pounds. {As; As I Pound, is to the Interest of Pound; fo is any Then { Principal or given Sum, to its Intereft. That is .. tr: p. ptr 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 ptr + p = a From whence the three following Theorems are eafily deduced. These four Theorems refolve all Questions about fimple Intereft. Queft. 1. What will 2561. 10s. amount to in 3 Years, 1 Quarter, 2 Months and 18 Days, at 6 per Cent. per Annum? Here is given p256.5, r=0.06, and t3.465993 Quere a per Theorem 1. For 3 Years 3 One Quarter=0.25 2 Monthso.16647 = 08333×2 |