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, d e, j; f g, &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 point p 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 Thus A C, A D, A E, &c. are the logarithms of o c, o d, o e, &c. respectively; and o a is the quantity whose logarithm is supposed equal to nothing.
We have here abstracted from numbers, that the doctrine may be the more gene ral ; but it is plain, that if A C, A D, A E, &c. he supposed, 1, 2, 3, &c. 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 2 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 0 p in creases proportionally, 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 to a greater is negative ; as this ratio, compounded with any other ratio, dimin ishes it. The ratio of any quantity A to unity, compounded with the ratio of unity to A, produces the ratio of A to A, or the ratio of equality ; and the measures of those two ratios destroy each other when added together ; so that when the one is considered as positive the other is to be considered as negative. By supposing the logarithms of quantities greater than o a (which is supposed to represent unity) to be positive, and the logarithms of quan tities less than it to be negative, the same rules serve for the operations by loga rithms, whether tnespantities be greater or less than o a. When o p increases proportionally, the motion of p is per petually accelerated ; for the spaces a c, c d, d e, &c. that are described by it in any equal times that continually succeed after each other, perpetually increase in the same proportion as the lines o a, o c, o d, &c. When the pointp moves from a towards o, and o p decreases proportion ally, the motion of p is perpetually re tarded ; for the spaces described by it in any equal times that continually succeed after each other, decrease in this case in the same proportion as o p decreases.
If the velocity of the point p be always as the distance o p, then will this line in crease or decrease in the manner sup posed by Lord Neper ; and the velocity of the point p being the fluxion of the line • p, will always vary in the same ratio as this quantity itself. This, we presume, will give a clear idea of the genesis, or nature of logarithms ; but for more of this doctrine, see Maclaurin's Fluxions.