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Logarithms

arithmetical, geometrical, progressionals, progressional, root, logarithm and pro

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LOGARITHMS, are the indexes OT ex ponents (mostly whole numbers and de cimal fractions, consisting of seven places of figures at least) of the powers or roots (chiefly broken) of a given number ; yet such indexes or exponents, that the seve ral powers or roots they express, are the natural numbers 1, 2, 3, 4, 5y &c. to 10 or 100000, &c. (as if the given number be 10, and its index be assumed 1.0000000, then the 0.0000000 root of 10, which is 1, will be the logarithm of 1; the 0.301036 root of 10, which is 2, will be the logarithm of 2; the 0.477121 root of 10, which is 3, will be the logarithm of 3 ; the 0.612060 root of 10, the logarithm of 4; the 1.041395 power of 10, the logarithm of 11 ; the 1.079181 power of 10, the logarithm of 12, &c.) being chiefly contrived for ease and expedition in performing of arithme tical opei ations in large numbers, and in trigonometrical calculations; but they have likewise been found of extensile service in the higher geometry, particularly in the method of fluxions. They are gene rally founded on this consideration, that if there be any row of geometrical pro portional numbers, as 1, 2, 4, 8, 16, 32, 64, 128, 256, &c. or 1, 10, 100, 1000, 10000, &c. And as many arithmetical progressional numbers adapted to them, or set over them, beginning with O.

0, 1, 2, 3, 4, 5, 6, 7, thus' C 1, 2, 4, 8, 16, 32, 64, 128, &c.

0, 1, 2, 3, 4, &c. ? or, 1, 10, 100, 1000, 10000, &c.5 Then will the sum of any two of these arithmetical progressionals, added toge ther, be that arithmetical progressional which answers to, or stands over the ge ometrical progressional, which is the pro duct of the two geometrical progression als,over which the two assumed arithme. tical progressionals stand ; again, if those arithmetical progressionals be subtracted from each other, the remainder will be the arithmetical progressional standing over that geometrical progressional, which is the quotient of the division of the two geometrical progressionals belong ing to the two first assumed arithmetical progressionals ; and the double, triple, &c. of any one of the arithmetical pro gressionals will be the arithmetical pro. gressional standing over the square, cube, &c. of that geometrical progression which the assumed arithmetical progressional stands over, as well as the one-half, one third, &c. of that arithmetical progres.

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 any one looks into the tables of logarithms, he will find, that these do not all run on in an arithmetical progression, nor the numbers they answer 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 " Harmonia Mensurarum," says, they are the numerical measures of ratios: but all these definitions convey but a very confhsed notion of logarithms. Mr. Maclaurin, in his "Treatise of Fluxion," 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 line. 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, de to d o, e f to e o, f g to f o, always in the same ratio of Q It to Q S : and sup pose the point P sets out from a, describ ing a c, c d, d c, e f, f g, 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.

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