AN ESSAY Logarithmical Arithmetick. CHA P. I. Of DEFINITIONS, and the Nature and Investigation of LOGARITHMS. I. L OGARITHMS (y and as) are Numbers fo contrived and adapted to other Numbers, that by their Addition, or Subtraction, the Product, or Quotient, of the Numbers to which they are adapted, may be found. This Definition expreffes the principal Defign and Use of thofe artificial Numbers, which Mathematicians distinguish by the Name of Logarithms. But here, (as it frequently happens in other Sciences,) a clear, accurate, etymological Senfe, cannot be underftood, till we have made a particular Enquiry into the Principles of the Science itself. 2. To proceed then, let us confider the Nature of a geometrical Progreffion, with the Indices fet over the respective Terms. B For Indices, o. For Example: Ï. 2. 3. 4. 5. 6. &c. Here it is evident, that if we add any two Indices together, their Sum will be the Index of that Number, which is equal to the Product of the Numbers whofe Indices are added together: Thus, e. g. The Indices 2 and 3 added together are = 5; the Numbers answering to the Indices 2 and 3, are 4 and 8; and 4×832 the Number answering to the Index 5. It also follows from the Nature of fuch Progreffions, that if we fubtract one Index from another, their Difference will be the Index of that Number or Term, which is equal to the Quotient of the two Numbers or Terms, correfponding to the Indices whose Differences we found: e. g. The Indices 642; the Numbers correfponding to these Indices are 64 and 16, and 64164 Number correfponding to the Index 2. the 3. Hence it follows, that (by Art. 1.) the Indices of a Series of Numbers in geometrical Progreffion, are Logarithms of the Terms in that Progreffion. 4. Hence, by the Nature of geometrical Progreffions, it follows, that, if the Logarithm of any Number be multiplied by the Index of the Power, the Product will be equal to the Logarithm of the Root when involved to the Height denoted by the Index. Thus, for Inftance: The Index or Loga rithm of 4 in the above Series 2, being multiplied by 36 the Index or Logarithm of 64; and 644 cubed = 4 X 4 X 4. 5. From hence alfo it follows, that the Index or Logarithm of any Number being divided by 2, the Quotient will be the Index or Logarithm of the fquare Root; but if divided by 3, the Index or Logarithm of the cube Root; if divided by 4, the Quotient will be the Index or Logarithm of the 4th Power, &c. . Example. Example. The Logarithm of 64 in the above Series is 6, which being divided by 3, the Quotient is 2, the Index or Logarithm of 4, which is the cube Root of 64; for 4 X 4 X 4 = 64. 6. It follows, from the Nature of geometrical Progreffions, that the Ratio of the firft, and any other Term, is compofed o fo many equal Ratios, as are expreffed by the Index of that other Term: e. g. In the above Series, the Ratio of the first and 7th Term is made up of 6 equal Ratios, for the Ratio of the ft to the 2d Term is as 1 to 2, and of the 2d to the 3d as 1 to 2, and of the 3d to the 4th as 1 to 2, of the 4th to the 5th as 1 to 2, of the 5th to the 6th as 1 to 2, and of the 6th to the 7th as 1 to 2;. the Ratio of the 1ft to the 7th is compofed of IXIXIXIXIXI to 2×2×2×2×2×2; i. e. as 1 to 64. Hence plainly appears the Propriety of the Term Logarithms, it fignifying, according to its Etymology, a Number of Ratios. 7. Logarithms may be of various Kinds; for inftead of the Series 1, 2, 4, &c. above, we may fubftitute any other; thus e. g. if we write the geometrical Series 1, 10, 100, &c. inftead of 1, 2, 4, &c. it will ftand thus: Indices or Logarithms o. 1. 2. 3: 4. &c. Terms or natural Nos. 1. 10. 100. 1000. 10000, &c. This is the Form of Brigs's Logarithms; which are those we fhall now make it our Bufinefs to difcourse on. The greatest Difficulty now remaining is to explain the Method made Ufe of by Mr. Brigs, for finding the Logarithms of the intermediate Terms, because the natural Numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, &c. are not in a geometrical Progreffion. 8. Before we proceed any farther, it may be proper to obferve, that, if we fuppofe a geometrical Series beginning from Unity, whofe common Multiplier is +m, the Series will be 1, I + m, 1 + m2, 1 + m3, 1+m, &c. Now if m be fuppofed B 2 pofed indefinitely little, it is evident that fome of the Terms will approach indefinitely near to the natural Numbers 2, 3, 4, 5, &c, and, confequently, it is poffible to find the natural Number anfwering to any Logarithm, very nearly, though not exactly true. 9. The Method of doing which may be thus explained. By the Definition of Logarithmns, if x and y represent any two Numbers, the Logarithm of Xy the Logarithm of x + the Logarithm of y; and therefore, (by Art. 5.) the Logarithm of the Log. +Log.y; Hence, the *arith 2 metical Mean of any two Logarithms is equal to the Logarithm of the geometrical Mean of the two natural Numbers; confequently, the Logarithm of any intermediate Number may be found, by finding the geometrical Mean between the two given Numbers, (whose Logarithms are given,) and the Logarithm anfwering thereto, (viz. the arithmetical Mean of the two given Logarithms :) Then, the geometrical Mean between the geometrical Mean just found and the nearest Extreme, and the Logarithm anfwering thereto. After this Manner we proceed, till we find a geométrical Mean so very nearly equal to the Number whofe Logarithm we want, as we shall think fufficiently near the Truth; then the Logarithm of that geometrical Mean is fufficiently near the Loga rithm required. 10. To illuftrate this, let it be required to find the Logarithm of Here we have the Logarithms of 1 and 10 given. 9. * Half the Sum of any two Numbers is called an arithmetical Mean; and the fquare Root of the Product of two Numbers iş called the geometrical Mean. (1,) ་ |