LOGARITHMS. 



sional, will be the geometrical progres- 

 sional answering to the square root, cube 

 root, &c. of the arithmetical progres- 

 sional over it ; and from hence arises the 

 following common, though imperfect de- 

 finition of logarithms ; viz. 



That they are so many arithmetical pro- 

 gressionals, answering to the same num- 

 ber of geometrical ones. Whereas, if 

 anyone looks into the tables of logarithms, 

 he will find, that these do not all run 

 on in an arithmetical progression, nor 

 the numbers they ajiswer to in a geome- 

 trical one ; these last being themselves 

 arithmetical progressionals. Dr. Wallis, 

 in his history of algebra, calls loga- 

 rithms the indexes of the ratios of num- 

 bers to one another. Dr. Halley, in the 

 Philosophical Transactions, Number 216, 

 says, they are the exponents of the ratios 

 of unity to numbers. So, also, Mr. Cotes, 

 in his " Hannonia Mensurarum," says, 

 they are the numerical measures of ratios: 

 but all these definitions convey but a 

 very confused notion of logarithms. Mr. 

 Maclaurin, in his "Treatise of Fluxions," 

 has explained the natural and genesis of 

 logarithms, agreeably to the notion of 

 their first inventor, Lord Neper. Loga- 

 rithms then, and the quantities to which 

 they correspond, may be supposed to be 

 generated by the motion of a point : and 

 if this point moves over equal spaces in 

 equal times, the line described by it in- 

 creases equally. 



Again, a line decreases proportionably, 

 when the point that moves over it des- 

 cribes such parts in equal times as are 

 always in the same constant ratio to the 

 lines from which they are subducted, or 

 to the distances of that point, at the be- 

 ginning of those lines, from a given term 

 in that lhie. In like manner, a line may 

 increase proportionably, if in equal times 

 the (moving point describes spaces pro- 

 portional to its distances from a certain 

 term at the beginning of each time. 

 Thus, in the first case, let a c (Plate IX. 

 Miscel. fig. 1 and 2.) be to a o, c d to c o, 

 d e to (I o, e f to e o,fg to/o, always in 

 the same ratio of Q R to Q S : and sup- 

 pose the point P sets out from fl, describ- 

 ing c, c d, d c, e /, fg, in equal parts 

 of the time ; and let the space described 

 by P in any given time, be always in the 

 same ratio to the distance of P from o at 

 the beginning of that time, then will the 

 right line a o decrease proportionally. 



In like manner, the line o a (fig. 3.) in- 

 creases proportionally, if the point p, in 

 equal times, describes spaces a c, c d, de, 

 e J> f ff> &-C. so that a c is to a o, c d to 

 c o, d e to d o, &.c. in a constant ratio. If 



we now suppose a point P describing the 

 line A G (fig. 4.) with an uniform motion, 

 while the pointy describes a line increas- 

 ing or decreasing proportionally, the line 

 A P, described by P, with this uniform 

 motion, in the same time that o a, by in- 

 creasing or decreasing proportionally, be- 

 comes equal to o p, is the logarithm of 

 op. Thus A C, A D, A E, &c. are the 

 logarithms of o c, o d, o e t &c. respectively; 

 and o a is the quantity whose logarithm is 

 supposed equal to nothing. 



We have here abstractedfrom numbers, 

 that the doctrine may be the more gene- 

 ral ; but it is plain, that if A C, A D, A E, 

 See. be supposed, 1, 2, 3, 8cc. in arithmetic 

 progression ; o c, o d, o e, &c. will be in 

 geometric progression ; and that the loga- 

 rithm of o a, which may be taken for 

 unity, is nothing. 



Lord Neper, in his first scheme of loga- 

 rithms, supposes, that while o p increases 

 or decreases proportionally, the uniform 

 motion of the point P, by which the loga- 

 rithm of o p is generated, is equal to the 

 velocity of p at a ; that is, at the term of 

 time when the logarithms begin to be 

 generated. Hence logarithms, formed 

 after this model, are called Neper's Loga- 

 rithms, and sometimes Natural Loga- 

 rithms. 



When a ratio is given, the point p de- 

 scribes the difference of the terms of the 

 ratio in the same time. When a ratio is 

 duplicate of another ratio, the point p de- 

 scribes the difference of the terms in a 

 double time. When a ratio is triplicate 

 of another, it describes the difference of 

 the terms in a triple time ; and so on. 

 Also, when a ratio is compounded of two 

 or more ratios, the point p describes the 

 difference of the terms of that ratio in a 

 time equal to the sum of the times, in 

 which it describes the difference of the 

 terms of the simple ratios of which it is 

 compounded. And what is here said of 

 the times of the motion of p when o p in- 

 creases proportionallv, is to be applied to 

 the spaces described'by P, in those times, 

 with its uniform motion. 



Hence the chief properties of loga- 

 rithms are deduced They are the mea- 

 sures of ratios. The excess of the loga- 

 rithm of the antecedent, above the loga- 

 rithm of the consequent, measures the 

 ratio of those terms. The measure of 

 the ratio of a greater quantity to a lesser 

 is positive ; as this ratio, compounded 

 with any other ratio, increases it. The 

 ratio of equality, compounded with any 

 other ratio, neither increases nor dimin- 

 ishes it ; and its measure is nothing. The 

 measure of the ratio of a lesser quantity" 



