are the indexes or exponents (mostly whole numbers and decimal 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 several powers or roots they express, are the natural numbers, 1, 2, 3, 4, 5, &c. to 10 or 100000, &c. (as, if the given number be 10, and its index be assumed 10000000, then the 0.000000 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 1.612060 root of 10, the logarithm of 4; the 1.041393 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 arithmetical operations in large numbers, and in trigonometrical calculations; but they have likewise been found of extensive service in the higher geometry, particularly in the method of fluxions. They are generally founded on this consideration, that if there be any row of geometrical proportional numbers, as 1, 2, 4, 8, 16, 32, 64, 128, 256, &c. or 1, 10, 100, 1000, 10000, &c. and as many arithmetical progressionals numbers adapted to them, or set over them, beginning with o,
\[ \begin{align*} &\{0, 1, 2, 3, 4, 5, 6, 7, \ldots\} \\ &\{1, 2, 4, 8, 16, 32, 64, 128, \ldots\} \end{align*} \]
or,
\[ \begin{align*} &\{0, 1, 2, 3, 4, \ldots\} \\ &\{1, 10, 100, 1000, 10000, \ldots\} \end{align*} \]
then will the sum of any two of these arithmetical progressionals, added together, be that arithmetical progression which answers to or stands over the geometrical progression, which is the product of the two geometrical progressionals over which the two assumed arithmetical progressionals stand: again, if those arithmetical progressionals be subtracted from each other, the remainder will be the arithmetical progression standing over that geometrical progression which is the quotient of the division of the two geometrical progressionals belonging to the two first assumed arithmetical progressionals; and the double, triple, &c. of any one of the arithmetical progressionals, will be the arithmetical progression standing over the square, cube, &c. of that geometrical progression which the assumed arithmetical progression stands over, as well as the \(\frac{1}{2}, \frac{1}{3}, \ldots\) of that arithmetical progression will be the geometrical progression answering to the square root, cube root, &c. of the arithmetical progression over it; and from hence arises the following common, though lame and imperfect definition of logarithms; viz.
"That they are so many arithmetical progressionals, answering to the same number 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 geometrical one; these last being themselves arithmetical progressionals. Dr Wallis, in his history of algebra, calls logarithms the indexes of the ratios of numbers to one another. Dr Halley, in the philosophical transactions, no 216, says, they are the exponents of the ratios of unity to numbers. So also Mr Cotes, in his Harmonia Mensurorum, 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 Napier. Logarithms 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 increases equally.
Again a line decreases proportionably, when the point that moves over it describes 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 beginning 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 proportional to its distances from a certain term at the beginning of each time. Thus, in the first case, let \(a c\) (Plate CIV. fig. 3.) be to \(a o, c d\) to \(c o, d e\) to \(d o, e f\) to \(e o, f g\) to \(f o\), always in the same ratio of \(Q R\) to \(Q S\); and suppose the point \(P\) sets out from \(a\), describing \(a c, c d, d e, e f, f g\), 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 proportionably.
In like manner, the line \(o a\), (ibid. no 3.) increases proportionally, if the point \(p\), in equal times, describes the spaces \(a c, c d, d e, f g, \ldots\) so that \(a c\) is to \(a o, c d\) to \(c o, d e\) to \(d o, \ldots\) in a constant ratio. If we now suppose a point \(P\) describing the line \(A G\) (ibid. no 4.) with an uniform motion, while the point \(p\) describes a line increasing or decreasing proportionally, the line \(A P\), described by \(P\), with this uniform motion; in the same time that \(o a\), by increasing or decreasing proportionally, becomes equal to \(o p\), is the logarithm of \(o p\): Thus \(A C, A D, A E, \ldots\) are the logarithms of \(o c, o d, o e, \ldots\) 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 general; but it is plain, that if \(A C, A D, A E, \ldots\) be supposed, 1, 2, 3, \ldots\) in arithmetical progression; \(o c, o d, o e, \ldots\) will be in geometric progression; and that the logarithm of \(o a\), which may be taken for unity, is nothing.
Lord Napier, in his first scheme of logarithms, supposes, that while \(o p\) increases or decreases proportionally, the uniform motion of the point \(P\), by which the logarithm 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. generated. Hence logarithms, formed after this model, are called Napier's Logarithms, and sometimes Natural Logarithms.
When a ratio is given, the point \( p \) describes the difference of the terms of the ratio in the same time. When a ratio is duplicate of another ratio, the point \( p \) describes 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 differences 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 \( op \) increases proportionally, is to be applied to the spaces described by \( P \), in those times, with its uniform motion.
Hence the chief properties of logarithms are deduced. They are the measures of ratios. The excess of the logarithm of the antecedent above the logarithm 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 diminishes 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, diminishes 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 \( aa \) (which is supposed to represent unity) to be positive, and the logarithms of quantities less than it to be negative, the same rules serve for the operations by logarithms, whether the quantities be greater or less than \( aa \). When \( op \) increases proportionally, the motion of \( p \) is perpetually accelerated; for the spaces \( ac, cd, de, \ldots \) that are described by it in any equal times that continually succeed after each other, perpetually increase in the same proportion as the lines \( oa, oc, od, \ldots \). When the point \( p \) moves from \( a \) towards \( o \), and \( op \) decreases proportionally, the motion of \( p \) is perpetually retarded; 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 \( op \) decreases.
If the velocity of the point \( p \) be always as the distance \( op \), then will this line increase or decrease in the manner supposed by lord Napier; and the velocity of the point \( p \) being the fluxion of the line \( op \), 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.
Construction of Logarithms.
The first makers of logarithms had in this a very laborious and difficult task to perform; they first made choice of their scale or system of logarithms, that is, what set of geometrical numbers should answer to such a set of geometrical ones, for this is entirely arbitrary; and they chose the decuple geometrical progressionals, \( 1, 2^{\text{th}}, 3^{\text{th}}, 4^{\text{th}}, 5^{\text{th}}, 6^{\text{th}}, 7^{\text{th}}, \ldots \); and the arithmetical one, \( 0, 1, 2, 3, \ldots \); as the most convenient after this they were to get the logarithms of all the intermediate numbers between \( 1 \) and \( 10 \), \( 10 \) and \( 100 \), \( 100 \) and \( 1000 \), \( 1000 \) and \( 10000 \), &c. But first of all they were to get the logarithms of the prime numbers \( 3, 5, 7, 11, 13, 17, 19, 23, \ldots \); and when these once had, it was easy to get those of the compound numbers made up of the prime ones, by the addition or subtraction of their logarithms.
In order to this, they found a mean proportion between \( 1 \) and \( 10 \), and its logarithm will be \( \frac{1}{2} \) that of \( 10 \); and so given, they found a mean proportional between the number first found and unity, which mean will be nearer to \( 1 \) than that before, and its logarithm will be \( \frac{1}{2} \) of the former logarithm, or \( \frac{1}{2} \) of that of \( 10 \); and having in this manner continually found a mean proportional between \( 1 \) and the last mean, and bisectioned the logarithms, they at length, after finding 54 such means, came to a number \( 00000000000001278191493200323442 \), so near to \( 1 \) as not to differ from it so much as... But since the way above hinted at, for finding the logarithms of the prime numbers is so intolerably laborious and troublesome, the more skilful mathematicians that came after the first inventors, employing their thoughts about abbreviating this method, had a vastly more easy and short way offered to them from the contemplation and mensuration of hyperbolic spaces contained between the portions of an asymptote, right lines perpendicular to it, and the curve of the hyperbola: for if ECN (Plate CIII. fig. 6 n° 1.) be an hyperbola, and AD, AQ the asymptotes, and AB, AP, AQ, &c. taken upon one of them, be represented by numbers, and the ordinates BC, PM, QN, &c. be drawn from the several points B, P, Q, &c. to the curve, then will the quadrilinear spaces BCMP, PMNQ, &c. viz. their numerical measures be the logarithms of the quotients of the division of AB by AP, AP by AQ, &c. since when AB, AP, AQ, &c. are continual proportionals, the said spaces are equal, as is demonstrated by several writers concerning conic sections.
**Hyperbola.**
Having said that these hyperbolic spaces, numerically expressed, may be taken for logarithms, we shall next give a specimen, from the great Sir Isaac Newton, of the method how to measure these spaces, and consequently of the construction of logarithms.
Let CA (ibid. n° 2.) = AF be = 1, and AB = Ab = x; then will \( \frac{1}{1+x} \) be = BD, and \( \frac{1}{1-x} \) = bd; and putting these expressions into series, it will be \( \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + \ldots \), &c. and \( \frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + x^5 + \ldots \), &c. and taking the fluents, we shall have the area AFDB = \( \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} + \ldots \), &c. and the area AFdb = \( \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} + \ldots \), &c. and the sum bd DB = \( \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} + \ldots \), &c.
Now if AB or ab be \( \frac{1}{x} = x \), CB being = 0.9, and CB = 1.1, by putting this value of x in the equations above, we shall have the area bd DB = 0.2006706954621511 for the terms of the series will stand as you see in this table:
| Term of the series | Value | |-------------------|-------| | first | 0.2006706954621511 | | second | 666666666666666666 | | third | 400000000000000000 | | fourth | 285714286 | | fifth | 2222222 | | sixth | 18182 | | seventh | 154 | | eighth | 1 |
If the parts Ad and AD of this area be added separately, and the lesser DA be taken from the greater dA, we shall have Ad - AD = \( \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \frac{x^5}{5} + \ldots \), &c. = 0.0100503358535014, for the terms reduced to decimals will stand thus:
| Term of the series | Value | |-------------------|-------| | first | 0.010000000000000000 | | second | 500000000000000000 | | third | 333333333333333333 | | fourth | 250000000000000000 | | fifth | 200000000000000000 | | sixth | 1667 | | seventh | 14 |
Now if this difference of the areas be added to, and subtracted from their sum before found, half the aggregate, viz. 0.1053605156578263 will be the greater area Ad, and half the remainder, viz. 0.0953101798043249, will be the lesser area AD.
By the same tables, these areas AD and Ad, will be obtained also when AB = Ab are supposed to be \( \frac{1}{10} \) or CB = 1.01, and CB = 0.99, if the numbers are but duly transferred to lower places, as
| Term of the series | Value | |-------------------|-------| | first | 0.02000006667066694 | | second | 666666666666666666 | | third | 400000000000000000 | | fourth | 28 |
Sum = 0.02000006667066694 = area bD.
Half the aggregate 0.0100503358535014 = Ad, and half the remainder, viz. 0.0099503308531681 = AD.
And so putting AB = Ab = \( \frac{1}{10} \), or CB = 1.01 and CB = 0.99, there will be obtained Ad = 0.010000000335835, and AD = 0.00999950013330835.
After the same manner, if AB = Ab, be = 0.2, or 0.02, or 0.002, these areas will arise.
Ad = 0.2231435513142097, and AD = 0.1823215576939546, or Ad = 0.0202027073175194, and AD = 0.1098026272961797, or Ad = 0.002002, and AD = 0.0001.
From these areas thus found, others may be easily had from addition and subtraction only. For since \( \frac{1.2}{0.8} \times \frac{1.2}{0.9} = 2 \), the sum of the areas belonging to the ratios \( \frac{1.2}{0.8} \) and \( \frac{1.2}{0.9} \) (that is, insisting upon the parts of the absciss 1.2, 0.8; and 1.2, 0.9) viz.
\[ \begin{align*} \text{AD} &= 0.18232, \\ \text{Ad} &= 0.10980, \\ \text{Ad} &= 0.02020, \\ \text{AD} &= 0.0001. \end{align*} \]
Sum = 0.28768, &c.
added thus,
\[ \begin{align*} \text{Ad} &= 0.40546, \\ \text{AD} &= 0.28768. \end{align*} \]
Total = 0.69314, &c. = the area of AFHG, when when CG is =2. Also, since \( \frac{1}{2} \times 2 = 3 \), the sum
\( 1.0986122, \) &c. of the areas belonging to \( \frac{1}{2} \) and 2,
will be the area of AFGH, when CG=3. Again, since
\( 2 \times 2 \)
\( 0.8 = 5 \), and \( 2 \times 5 = 10 \); by adding \( AD = 0.2231, \) &c.
\( AD = 0.1823, \) &c. and \( AD = 0.1053, \) &c. together, their
sum is 0.5108, &c. and this added to 1.0986, &c.
the area of AFGH, when CG=3. You will have
\( 1.6093379124341004 = \text{AFGH}, \) when CG is 5; and
adding that of 2 to this, gives \( 2.3025850929940457 = \text{AFGH}, \) when CG is equal to 10: and since \( 10 \times 10 = 100; \) and \( 10 \times 100 = 1000; \) and \( \sqrt[5]{5} \times 10 \times 0.98 = 7, \) and
\( 10 \times 1.1 = 11, \) and \( \frac{1000 \times 1.001}{7 \times 11} = 13, \) and \( \frac{1000 \times 0.998}{2} = 499; \)
it is plain that the area AFGH may be found by
the composition of the areas found before, when CG=100,
1000, or any other of the numbers above mentioned;
and all these areas are the hyperbolic logarithms of those
several numbers.
Having thus obtained the hyperbolic logarithms of the
numbers 10, 0.98, 0.99, 1.01, 1.02; if the logarithms
of the four last of them be divided by the hyperbolic lo-
garithm 2.3025850, &c. of 10, and the index 2, be added
or, which is the same thing, if it be multiplied by
its reciprocal 0.4342944819032518, the value of the
subtangent of the logarithmic curve, to which Briggs's lo-
garithms are adapted, we shall have the true tabular loga-
rithms of 98, 99, 100, 101, 102. These are to be interpo-
lated by ten intervals, and then we shall have the logarithms
of all the numbers between 98 and 1020; and all between
980 and 1000, being again interpolated by ten intervals,
the table will be as it were constructed. Then from these
we are to get the logarithms of all the prime numbers,
and their multiples less than 100, which may be done by
addition and subtraction only: for \( \frac{\sqrt[5]{84 \times 1020}}{9945} = 2; \)
\( \frac{8 \times 9963}{984} = 3; \frac{10}{2} = 5; \frac{\sqrt[5]{98}}{2} = 7; \frac{99}{9} = 11; \frac{1001}{7 \times 11} = 13; \)
\( \frac{102}{6} = 17; \frac{988}{4 \times 13} = 19; \frac{9936}{16 \times 27} = 23; \frac{986}{2 \times 17} = 29; \frac{992}{32} = 31; \frac{999}{27} = 37; \frac{984}{24} = 41; \frac{989}{23} = 43; \frac{987}{21} = 47; \)
\( \frac{991}{11 \times 17} = 53; \frac{9971}{13 \times 13} = 59; \frac{988}{2 \times 81} = 61; \frac{9949}{3 \times 49} = 67; \)
\( \frac{994}{14 \times 71} = 71; \frac{9928}{8 \times 17} = 73; \frac{9954}{7 \times 18} = 79; \frac{996}{12} = 83; \frac{9968}{7 \times 16} = 89; \frac{9894}{6 \times 17} = 97; \)
and thus having the logarithms
of all the numbers less than 100, you have nothing to
do but interpolate the several times, through ten inter-
vals.
Now the void places may be filled up by the following
theorem. Let \( n \) be a number, whose logarithm is want-
ed; let \( x \) be the difference between that and the two
nearest numbers, equally distant on each side, whose lo-
garithms are already found; and let \( d \) be half the differ-
ence of their logarithms; then the required logarithm
of the number \( n, \) will be had by adding \( d + \frac{dx}{2n} + \frac{dx^3}{12n^3}, \)
&c. to the logarithm of the lesser number; for if the
numbers are represented by \( CP, \) CG, CP, (ibid. no. 2.)
and the ordinates \( PS, PQ, \) be raised; if \( n \) be wrote for
CG, and \( x \) for GP, or Gp, the area \( PSQP, \) or \( \frac{2x}{n} + \frac{x^3}{2n^3} + \frac{x^5}{3n^5}, \) &c. will be to the area \( PSHG, \) as the difference
between the logarithms of the extreme numbers, or \( 2d, \) is
to the difference between the logarithms of the lesser, and
of the middle one; which, therefore, will be
\( \frac{dx}{n} + \frac{dx^3}{2n^3} + \frac{dx^5}{3n^5}, \) &c.
The two first terms \( d + \frac{dx}{2n} \) of this series, being suffi-
cient for the construction of a canon of logarithms, even
to 14 places of figures, provided the number, whose lo-
garithm is to be found, be less than 1000; which cannot
be very troublesome, because \( x \) is either 1 or 2: yet it is
not necessary to interpolate all the places by help of this
rule, since the logarithms of numbers, which are produ-
ced by the multiplication or division of the number last
found, may be obtained by the numbers whose logarithms
were had before, by the addition or subtraction of their
logarithms. Moreover, by the difference of their loga-
rithms, and by their second and third differences, if ne-
cessary, the void places may be supplied more expediti-
ously; the rule afore-going being to be applied only where
the continuation of some full places is wanted, in order
to obtain these differences.
By the same method rules may be found for the inter-
calation of logarithms, when of three numbers the loga-
rithm of the lesser and of the middle number are given,
or of the middle number and the greater; and this al-
though the numbers should not be in arithmetical pro-
gression. Also by pursuing the steps of this method,
rules may be easily discovered for the construction of ar-
tificial sines and tangents, without the help of the natural
tables. Thus far the great Newton, who says, in one
of his letters to Mr Leibnitz, that he was so much de-
lighted with the construction of logarithms, at his first
setting out in these studies, that he was ashamed to tell
to how many places of figures he had carried them at that
time: and this was before the year 1666; because, he
says, the plague made him lay aside these studies, and
think of other things.
Dr. Keil, in his Treatise of Logarithms, at the end
of his Commandine's Euclid, gives a series, by means of
which may be found easily and expeditiously the loga-
rithms of large numbers. Thus, let \( z \) be an odd number,
whose logarithm is sought: then shall the numbers \( z-1 \)
and \( z+1 \) be even, and accordingly their logarithms, and
the difference of the logarithms will be had, which let be
called \( y. \) Therefore, also the logarithm of a number,
which is a geometrical mean between \( z-1 \) and \( z+1, \)
will LOG
will be given, viz., equal to half the sum of the logarithms.
Now the series \( \frac{1}{4z} + \frac{1}{24z^3} + \frac{181}{15120z^7} + \frac{25200z^9}{13} \), etc., shall be equal to the logarithm of the ratio, which the geometrical mean between the numbers \( z-1 \) and \( z+1 \), has to the arithmetical mean, viz., to the number \( z \). If the number exceeds 1000, the first term of the series, viz., \( \frac{y}{4z} \), is sufficient for producing the logarithm to 13 or 14 places of figures, and the second term will give the logarithm to 20 places of figures. But if \( z \) be greater than 0.000000000542813; and if the logarithm of the geometrical mean, viz., \( \frac{y}{4z} \), is added to the quotient, the sum will be
\[ 4.301051709302416 = \text{the logarithm of } 20001. \]
Wherefore it is manifest that to have the logarithm to 14 places of figures, there is no necessity of continuing out the quotient beyond 6 places of figures. But if you have a mind to have the logarithm to 10 places of figures only, the two first figures are enough. And if the logarithms of the numbers above 20000 are to be found by this way, the labour of doing them will mostly consist in setting down the numbers. This series is easily deduced from the consideration of the hyperbolic spaces aforesaid. The first figure of every logarithm towards the left hand, which is separated from the rest by a point, is called the index of that logarithm; because it points out the highest or remotest place of that number from the place of unity in the infinite scale of proportions towards the left hand: thus, if the index of the logarithm be 1, it shews that its highest place towards the left hand is the tenth place from unity; and therefore all logarithms which have 1 for their index, will be found between the tenth and hundredth place, in the order of numbers. And for the same reason all logarithms which have 2 for their index, will be found between the hundredth and thousandth place, in the order of numbers, etc. Whence universally the index or characteristic of any logarithm is always less by one than the number of figures in whole numbers, which answer to the given logarithm; and, in decimals, the index is negative.
As all systems of logarithms whatever, are composed of similar quantities, it will be easy to form, from any system of logarithms, another system in any given ratio; and consequently to reduce one table of logarithms into another of any given form. For as any one logarithm in the given form, is to its correspondent logarithm in another form; so is any other logarithm in the given form, to its correspondent logarithm in the required form; and hence we may reduce the logarithms of lord Napier into the form of Briggs's, and contrariwise. For as 2.302585092, etc., lord Napier's logarithm of 10, is to 1.0000000000, Mr Briggs's logarithm of 10; so is any other logarithm in lord Napier's form, to the correspondent tabular logarithm in Mr Briggs's form; and because the two first numbers constantly remain the same; if lord Napier's logarithm of any one number be divided by 2.302585, etc., or multiplied by 4342944, etc., the ratio of 1.00000, etc., to 2.30258, etc., as is found by dividing 1.00000, etc., by 2.30258, etc., the quotient in the former, and the product in the latter, will give the correspondent logarithm 10000, the first term will exhibit the logarithm to 13 places of figures; end so this series is of great use in filling up the chilads omitted by Mr Briggs. For example, it is required to find the logarithm of 20001; the logarithm of 20000 is the same as the logarithm of 2, with the index 4 prefixed to it; and the difference of the logarithms of 20000 and 20001, is the same as the difference of the logarithms of the numbers 10000 and 10001, viz., 0.0000434272, etc. And if this difference be divided by 4z, or 80004, the quotient \( \frac{y}{4z} \) shall be
in Briggs's form, and the contrary. And, after the same manner, the ratio of natural logarithms to that of Briggs's, will be found=868538963806.
The use and application of Logarithms.
It is evident, from what has been said of the construction of logarithms, that addition of logarithms must be the same thing as multiplication in common arithmetick; and subtraction in logarithms the same as division: therefore, in multiplication by logarithms, add the logarithms of the multiplicand and multiplier together, their sum is the logarithm of the product.
Example. Multiplicand 8.5 0.1294189 Multiplier 10 1.0000000
Product 85 1.9294189
And in division, subtract the logarithm of the divisor from the logarithm of the dividend, the remainder is the logarithm of the quotient.
Example. Dividend 97128 3.9873444 Divisor 456 2.6589648
Quotient 21.3 1.3283796
To find the complement of a Logarithm.
Begin at the left hand, and write down what each figure wants of 9, only what the last significant figure wants of 10; so the complement of the logarithm of 456, viz., 2.6589648, is 7.3410352.
In the rule of three. Add the logarithms of the second and third terms together, and from the sum subtract the logarithm of the first, the remainder is the logarithm of the fourth. Or, instead of subtracting a logarithm, add its complement, and the result will be the same.
To raise powers by Logarithms.
Multiply the logarithm of the number given, by the index of the power required, the product will be the logarithm of the power sought.
Example. Let the cube of 32 be required by logarithms. The logarithm of 32=1.5051500, which multiplied by 3, is 4.5154500, the logarithm of 32768, the cube of 32. But in raising powers, viz., squaring, cu-
bing, To find mean proportionals between any two numbers.
Subtract the logarithm of the least term from the logarithm of the greatest, and divide the remainder by a number more by one than the number of means desired; then add the quotient to the logarithm of the least term (or subtract it from the logarithm of the greatest) continually, and it will give the logarithms of all the mean proportionals required.
Example. Let three mean proportionals be sought, between 106 and 100.
Logarithm of 106 = 2.0253058 Logarithm of 100 = 2.0000000
Divide by 4)0.0253059(0.0063264.75
Logarithm of the least term 100 added 2.0000000
Logarithm of the first mean 101.4673846 2.0063264.75 Logarithm of the second mean 102.9563014 2.0126529.5 Logarithm of the third mean 104.4670483 2.0189794.25 Logarithm of the greatest term 106 2.0253059.