umber to the Log. ex less fat: 25 dition and subtraction of Lograr un ser s the multiplication sur bay : That is, the Log of ar butter. a will be the Log ofte Freurs inte be expressed in words as follows: From number sabact the Leg of the lesser, torresponding to the Remainder Then, if Lare to be added together, add to the num hey are to be subtracted, subtract 1 from umber thus increased or diminished addOver original number, will give the Log See required. tract that from the next greater Log. contained in the Table; multiply the difference by the remaining figure or figures in the given Number, and from the Product cut off as many figures from the Right hand as remain in the given Number; add the figure or figures standing at the Left hand to the Log. of the three first figures, and the Sum will be the Log. required, to which prefix the proper Index. EXAMPLES. Note. This is also the Log. of 762.4 or 76.24, &c. varying the Index according to the preceding directions. 2. Required the Logarithm of 541.25 Log. of 542 541 Difference .73400 .73320 80 To find the nearest Number corresponding to any Logarithm for more than three places of figures. Find the Log. next less than the given one, and take the differ ence between that and the given one; also take the difference between the next greater and the next less Log. than the given one; divide the former difference by the latter, according to the Rule in Divison of Decimals; add the Quotient to the number answering to the Log. next less than the given one, and will have the required Number; whether a whole or a mixed Number will be determined by the Index. EXAMPLES. 1. Required the Number to the Logarithm 3.88218 Given Log. Next less Difference .88195 Next less you .88252 .88195 57 57)23.0(4 228 2 The Number to the Log. next less than the given one is 7620 Because the Index is 3; to this add 4 and it makes 7624 the required Number. 2. Required the Number to the Logarithm 2.73340 Next greater Log. Difference 80)20.00(25 160 400 400 The Number to the Log. next less than the given one is 541', to this add the figures in the preceding Quotient, which are known to be Decimals from the Index of the given Log. and the required Number will be 541.25 The addition and subtraction of Logarithms answers the same purpose as the multiplication and division of their corresponding Numbers: That is, the Log. of any two Numbers being added, their Sum will be the Log of the Product of those Numbers; and the Log. of one Number being subtracted from the Log. of another Number, the Remainder will be the Log. of the Quotient of one of those Numbers divided by the other. Again, the Log. of any Number being doubled will produce the Log. of the Square of that Number; and one half the Log. of any Number is the Log. of the Square Root of that Number. To perform Addition or Subtraction by Logarithms. The following Theorems for adding and subtracting by Logarithms were invented by Mr. EBENEZER R. WHITE of DANBURY, and by him communicated to the Compiler. Though in common cases, they may not be particularly useful, yet in the solution of many Mathematical Questions they will greatly abridge the numerical operation. They are therefore here inserted. = Let a greater number to be added or subtracted. b= lesser These Theorems may be expressed in words as follows: From the Log. of the greater number subtract the Log. of the lesser, and find the number corresponding to the Remainder: Then, if the original numbers are to be added together, add 1 to the number last found; but if they are to be subtracted, subtract 1 from it; and the Log. of the number thus increased or diminished added to the Log. of the lesser original number, will give the Log. of the Sum or Difference required. Of the TABLE of LOGARITHMIC or ARTIFICIAL SINES, TANGENTS and SECANTS. To find the Logarithmic Sine, &c. for any number of Degrees and Minutes, within the Compass of the Table. If the Degrees be less than 45, look for them at the top of the Columns, and under Sine, Tangent or Secant, whichever is wanted, and for the Minutes at the left hand; but if more than 45, look for the Degrees at the Bottom over Sine, &c. and for the Minutes at the Right hand; under or over the Degrees and against the Minutes will be the required Log. Sine, &c. To find the Degrees and Minutes corresponding to a given Logarithmic Sine, &c. Look in the proper Column for the nearest Log. to the given one; and the degrees and Minutes standing over or under and against it, are those required. Note, When the Log. Sine, &c. for more than 90° is required, subtract the given number of Degrees from 180° and make use of the Remainder. It will be observed that this Table is calculated only for every 5 Minutes. This was thought sufficient for Surveyors, as few Compasses will take a Course to greater exactness. If however a Question is to be solved where greater accuracy is required, work by natural Sines. Or, The Log. Sine, &c. for any Minute may be found as follows: Look in the Table for the Log. of the nearest number of Minutes greater than the given one, and from this substract the next less Log. contained in the Table: Then say, As 5 Minutes, Is to this difference; So is the excess of the given Minutes above 5, 10, 15, 20, 25, &c; To a fourth number, which add to the Log. of the Minutes next less than the given number, and the sum will be the Log. required. To find the nearest Minutes corresponding to a given Logarithmic Sine, &c. Look in the Table, in the proper Column, for the Log. next 'less than the given one, and take the difference between that and the given one; also take the difference between the next greater and the next less Log. than the given one; Then say, As the latter difference; Is to 5 Minutes; So is the former difference; To the number of Minutes to be added to the Minutes of the Log. EXAMPLE. Required the Degrees and Minutes corresponding to the Loga, rithmic Tangent 9.73597. The Degrees and Minutes for the Log. next less than the given one are 28° 30', to which add 4' and it makes 28° 34' Note. As after the most careful attention of the Printers, some figures in the Table may be wrong; and as some may be so blurred as to be illegible, let it be observed, that the Sines and Co-Secants, the Co-Sines and Secants, and the Tangents and Co-Tangents, standing against each other respectively, being added together, will amount to 20.00000, if the Tables are accurate. Thus against 28° 20′ the Sine 9.67633 added to the Co-Secant 10.32367 their Sum is 20.00000; so also is the Sum of the Co-Sine 9.94458 and the Secant 10. 05542, and likewise the sum of the Tangent 9.73175 and the Co-Tangent 10.26825. An error may consequently be easily detected, or any defaced figure be supplied. To calculate the Northing or Southing, &c. for any Course and Distance by Logarithms. This is done by the first CASE of RIGHT ANGLED TRIGONOMETRY, as follows: Find the Log. Sine and Co-Sine of the Course; to each of these add the Log. of the Distance; subtract Radius or 10.00000 from their Sums, and the Remainders will be the Log. of the required Latitude and Departure. Note. When the Angle is very small or very large, and the Distance short, the sum of the Log. Sine or Co-Sine and the Log. of the Distance may be less than 10.00000 or Radius, which cannot therefore be subtracted. In such cases look for the Log. without regard to the Index, and the corresponding Number will be a Decimal, the first Figure of which will be Tenths if the Index be 9, and Hundredths if the Index be 8. Bb |