that to be fubtracted is lefs than that from which it is to be taken, find the Difference of their fractional Parts, to which prefix the Difference of their Characteristicks, with the affirmative Sign, if the Characteristick of the Logarithm to be fubtracted is the greater; otherwife it must have a negative Sign. But if the fractional Part of the Logarithm to be fubtracted, be greater than the fractional Part of the Logarithm from which it is to be taken, add 1 to the fractional Part of this laft, that Subtraction may be made; and then, (to continue the Equality, add 1 to the Character iftick of the Minorand, which Characteristick being negative, is done by Subtraction, viz. fubtract from the Characteristick of the Minorand; then the Difference between the Characteristick of the Minorand fo diminished, and the Characteristick of the Subducend, will give the required Characteristick; which must be affirmative, if the Characteristick of the Minorand, after being diminished as above, is greater than that of the Subducend: Otherwife negative. 31. Before we proceed to the Application of Logarithms, it will be proper to fhew how to find the natural Number anfwering to any given Logarithm, by Help of a Table of Logarithms. And as this is only the Reverse of Article 22, it will be fufficient, to hint, ift, That the Index, or Characteristick, of the Logarithm, if affirmative, will fhew how many integral Places are in the required Number: And if negative, in what Place of Decimals the first, or fignificant Figure, ftands. Hence, if the Logarithm can be found in the Table, the answering Number is found by a bare Inspection: But if it cannot be exactly found, find the next greater, and next leffer, and fay, As the Difference of these two Logarithms : the Difference of the anfwering Numbers :: the Difference between the given Logarithm and the nearest tabular Logarithm: a fourth Number; which added to, or fubtracted from, the natural Number answering to the neareft tabular Logarithm, (according (according as that Logarithm was lefs or greater than' the given one,) will give the required Number very near the Truth. Example. Let it be required to find the natural Number answering to the Logarithm 5.564882. This in the Tables gives the next leffer and greater 55.5647847 The answering ƒ 367100 Logarithms, 25.564903 $ Numbers 1367200 Their Diff. .000119 And 5.564903 As .000119 5.564882.000021. 100 Hence, 100 :: .000021; or, which is the fame in Effect, As 119: 100: 21: 17.7 nearly. Hence, 367200-17.7367182.3, nearly the required Number. 32. CHA P. II. MUTTIPLICATION by LOGARITHMS. WE E now come to fhew the Application of Logarithms, having explained their Theory at large; and the firft Thing is to perform Multiplication by them; which is done by adding the Logarithms of the Factors together, as directed in' Art. 23, 24, and 25. 33. Example 1. Multiply 14 by 4. Log. of 14 is 1.1461280 { 4 is 0.6020600' Their Sum 1.74818880 anfwers in the Table to 56, the required Product. 34. Example 2. Multiply 203 by 0.25. Log. of 203 is 2.3074960 Log. of 0.25 is 1.3979400 This Log. 1.7054360 anfwers in the Table to 35. Example 3. Multiply 7.677 by . 1.50.75. From 1.5841615 LIN Subtract Log. of 8 = 0.9030900 Gives Log. of 4.798 nearly, 0.6810715 Or thus: Put into Decimals, viz. = .625. Then, Løg 7.677 0.8851915 / Log. of or of .625 = 1.7958800 Sum is 0.6810715 as above. 36. Example 4. Multiply 0.55 by .023. Log. .55 is 1.7403627 Log..023 is 2.3617278 Log. of .01265 2.1020905 37. Example 5. Multiply 80 by .003. Log. 80 1.9030894 Log. .0033.4771212 Log..241.3802116 38. Example 6. Multiply, 10, 80, 72, .5, .4, I and .12, together. 39. CHA P. III, DIVISION by LOGARITHMS. TH HE Subtraction of Logarithms answering to the Divifion of the natural Numbers, from the Logarithm of the Dividend fubtract the Logarithm of the Divifor, by the Rules in Art. 26, 27, 28, 29, 30, and the Remainder will be the Logarithm of the required Quotient. 40. Example 1. Divide 56 by 4. Log. 56 1.7481880 42. Example 3. Divide 50.75 by .25. Logarithm of 59.75 1.7054360 Log. of .251.3979400 Log. of 203 2.3074960 44. Example 5. Divide .75 by 100. Log. of 75.8750613 Log, of .0075 = 3.87506132 45. Example 6. Divide 24 by 80. Log. of 241.3802112 Log. of .0033.4771212 46. Example 7. Divide .75 by .25, Log. of .75 = 1.8750613 |