LOGARITHMS

LOGARITHMS, 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 1.000000, 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 progression numbers adapted to them, or set over them, beginning with 0,

thus,

or,

then will the sum of any two of these arithmetical progression numbers, added together, be that arithmetical progression which answers to or stands over the geometrical progression, which is the product of the two geometrical progression numbers over which the two assumed

arithmetical progression numbers stand: again, if those arithmetical progression numbers 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 progression numbers belonging to the two first assumed arithmetical progression numbers; and the double, triple, &c. of any one of the arithmetical progression numbers, 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 $\backslash frac\{1\}\{2\}$, $\backslash frac\{1\}\{3\}$, &c. 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, tho' lame and imperfect definition of logarithms, viz. "That they are so many arithmetical progression numbers, 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 progression numbers. 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, n^o 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 confused notion of logarithms. Mr MacLaurin, in his Treatise of Fluxions, has explained the nature 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 subtracted, 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 Plate CLXII. the first case, let $ac$ be to $ae$, $cd$ to $ce$, $de$ to $de$, $ef$ to $eg$, $fg$ to $fh$, always in the same ratio of QR to QS, fig. 3. and suppose the point P sets out from $a$, describing $ac$, $cd$, $de$, $ef$, $fg$, 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 $a$ at the beginning of that time; then will the right line $ao$ decrease proportionably.

In like manner, the line $oa$, (ibid. n^o 3.) increases proportionably, if the point $p$, in equal times, describes the spaces $ac$, $cd$, $de$, $fg$, &c. so that $ac$ is to $ae$, $cd$ to $ce$, $de$ to $de$, &c. in a constant ratio. If we now suppose a point P describing the line AG (ibid. n^o 4.) with a uniform motion, while the point $p$ describes a line increasing or decreasing proportionally, the line AP, described by P, with this uniform motion, in the same time that $oa$, by increasing or decreasing proportionally, becomes equal to $op$, is the logarithm of $op$. Thus AC, AD, AE, &c. are the logarithms of

Logarithms of $oc,\; od,\; oe,\; \backslash dots$ respectively; and $oa$ 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 $AC,\; AD,\; AE,\; \backslash dots$ be supposed $1,\; 2,\; 3,\; \backslash dots$ in arithmetic progression; $oc,\; od,\; oe,\; \backslash dots$ will be in geometric progression; and that the logarithm of $oa$, which may be taken for unity, is nothing.

Lord Napier, in his first scheme of logarithms, supposes, that while $op$ increases or decreases proportionally, the uniform motion of the point $P$, by which the logarithm of $op$ 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 Naper'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 $oa$ (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 $oa$. When $op$ increases proportionally, the motion of $p$ is perpetually accelerated; for the spaces $ac,\; ed,\; de,\; \backslash dots$ that are described by it in any equal times that continually succeed after each other, perpetually increase in the same proportion as the lines $oa,\; oc,\; od,\; \backslash dots$ 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 al-

ways vary in the same ratio as this quantity itself. Logarithms. 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.

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 arithmetical progressionals should answer to such a set of geometrical ones, for this is entirely arbitrary; and they chose the decuple geometrical progressionals, $1,\; 10,\; 100,\; 1000,\; 10000,\; \backslash dots$ and the arithmetical one, $0,\; 1,\; 2,\; 3,\; 4,\; \backslash dots$ or $0,\; 000000,\; 1,000000,\; 2,000000,\; 3,000000,\; 4,000000,\; \backslash dots$ 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 $3,\; 5,\; 7,\; 11,\; 13,\; 17,\; 19,\; 23,\; \backslash dots$ and when these were 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 $\backslash frac\{1\}\{2\}$ that of $10$; and so given, then they found a mean proportional between the number first found and unity, which mean will be nearer to one than that before, and its logarithm will be $\backslash frac\{1\}\{4\}$ of the former logarithm, or $\backslash frac\{1\}\{4\}$ of that of $10$; and having in this manner continually found a mean proportional between $1$ and the last mean, and bisected the logarithms, they at length, after finding $54$ such means, came to a number $1.0000000000000001278191493200323442$, so near to $1$ as not to differ from it so much as $100000000000000000000$th part, and found its logarithm to be $0.000000000000000000005551115123125782702$, and $0.0000000000000000000012781914932003235$ to be the difference whereby $1$ exceeds the number of roots or mean proportionals found by extraction; and then, by means of these numbers, they found the logarithms of any other numbers whatsoever; and that after the following manner: Between a given number whose logarithm is wanted, and $1$, they found a mean proportional, as above, until at length a number (mixed) be found, such a small matter above $1$, as to have $1$ and $15$ cyphers after it, which are followed by the same number of significant figures; then they said, As the last number mentioned above is to the mean proportional thus found, so is the logarithm above, viz. $0.000000000000000000005551115123125782702$, to the logarithm of the mean proportional number, such a small matter exceeding $1$ as but now mentioned; and this logarithm being as often doubled as the number of mean proportionals (formed to get that number) will be the logarithm of the given number. And this was the method Mr. Briggs took to make the logarithms. But if they are to be made to only seven places of figures, which are enough for common use, they had only occasion to find $25$ mean proportionals, or, which is the same thing, to extract the $100000000000000000000$th root of $10$. Now having the logarithms of $3,\; 5$, and $7$, they easily got those of $2,\; 4,\; 6,\; 8$, and $9$; for since $3\; \backslash times\; 2\; =\; 6$, the logarithm of $2$ will be the difference of the logarithms of $6$ and $3$, the logarithm of $4$ will be two times the logarithm of $2$, the logarithm of $6$ will be the sum of the logarithm of $2$ and $3$, and the

Logarithms logarithm of 9 double the logarithm of 3. So, also having found the logarithms of 13, 17, and 19, and also of 23 and 29, they did easily get those of all the numbers between 10 and 30, by addition and subtraction only; and so having found the logarithms of other prime numbers, they got those of other numbers compounded of them.

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 CLXI. fig. 6. n^o 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 proportions, the said spaces are equal, as is demonstrated by several writers concerning conic sections.

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 the logarithms.

Let CA (ibid. n^o 2.) = AF be = 1, and AB = Ab = x; then will $\backslash frac\{1\}\{1+x\}$ be = BD, and $\backslash frac\{1\}\{1-x\}$ = bd; and putting

these expressions into series, it will be $\backslash frac\{1\}\{1+x\}\; =\; 1\; -\; x\; +\; x^2$

$-x^3\; +\; x^4\; -\; x^5$, &c. and $\backslash frac\{1\}\{1-x\}\; =\; 1\; +\; x\; +\; x^2\; +\; x^3\; +\; x^4\; +\; x^5$, &c.

and $\backslash frac\{x\}\{1+x\}\; =\; x\; -\; xx\; +\; x^2x\; -\; x^3x\; +\; x^4x\; -\; x^5x$, &c. and $\backslash frac\{x\}\{1-x\}\; =\; x\; +\; xx\; +\; x^2x\; +\; x^3x\; +\; x^4x\; +\; x^5x$, &c. and taking the fluents, we shall have the area AFDB = $x\; -\; \backslash frac\{xx\}\{2\}\; +\; \backslash frac\{x^3\}\{3\}\; -\; \backslash frac\{x^5\}\{5\}\; +\; \backslash frac\{x^7\}\{7\}\; -\; \backslash frac\{x^9\}\{9\}$, &c. and the area AFdb = $x\; +\; \backslash frac\{xx\}\{2\}\; +\; \backslash frac\{x^3\}\{3\}\; +\; \backslash frac\{x^5\}\{5\}\; +\; \backslash frac\{x^7\}\{7\}\; +\; \backslash frac\{x^9\}\{9\}$, &c.

Now, if AB or ab be $\backslash frac\{1\}\{10\}\; =\; x$, 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.

0.2006706954621511

If the parts Ad and AD of this area be added separate. Logarithmically, and the lesser DA be taken from the greater dA, we shall have $Ad\; -\; AD\; =\; x^2\; +\; \backslash frac\{x^4\}\{2\}\; +\; \backslash frac\{x^6\}\{3\}\; +\; \backslash frac\{x^8\}\{4\}\; +\; \backslash dots$ = 0.0100503358535014, for the terms reduced to decimals will stand thus:

0.0100503358535014

Now if this difference of the areas be added to, or 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 $\backslash frac\{1\}\{100\}$ or CB = 1.01, and Cb = 0.99, if the numbers are but duly transferred to lower places, as

Sum = 0.0200006667066694 = area bB.

0.0001000050003333 = area Ad - AD.

Half the aggregate 0.0100503358535014 = Ad, and half the remainder, viz. 0.0099503308531681 = AD.

And so putting AB = Ab = $\backslash frac\{1\}\{100\}$, or CB = 1.001 and Cb = 0.999, we obtain Ad = 0.0100050003335835, and AD = 0.009950013330835.

After the same manner, if AB = Ab, be = 0.2, or 0.02, or 0.002, these areas will arise.

From these areas thus found, others may be easily had from addition and subtraction only. For since $\backslash frac\{1.2\}\{0.8\}\; \backslash times\; \backslash frac\{1.2\}\{0.9\}$

= 2, the sum of the areas belonging to the ratios $\backslash frac\{1.2\}\{0.8\}$ and $\backslash frac\{1.2\}\{0.9\}$ (that is, insisting upon the parts of the abscissa 1.2, 0.8; and 1.2, 0.9) viz.

Sum = 0.28768, &c.

Total = 0.69314, &c. = the area of AFHG, when CG is = 2. Also, since $\backslash frac\{1.2\}\{0.8\}\; \backslash times\; 2\; =\; 3$, the sum

1.0986122,

1.0986122, &c. of the areas belonging to $\backslash frac\{1.2\}\{0.8\}$, and 2, will be the area of AFGH, when CG=3. Again, since $\backslash frac\{2\; \backslash times\; 2\}\{0.8\}\; =\; 5$, and $2\; \backslash times\; 5\; =\; 10$; 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 = AFGH, when CG is 5; and adding that of 2 to this, gives 2.3025850929940457 = AFGH, when CG is equal to 10; and since $10\; \backslash times\; 10\; =\; 100$, and $10\; \backslash times\; 100\; =\; 1000$, and $\backslash sqrt\{5\; \backslash times\; 10\; \backslash times\; 0.98\}\; =\; 7$, and $10\; \backslash times\; 1.1\; =\; 11$, and $\backslash frac\{1000\; \backslash times\; 1.091\}\{7\; \backslash times\; 11\}\; =\; 13$, and $\backslash frac\{1000\; \backslash times\; 0.998\}\{2\}\; =\; 499$, 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 logarithm 2.3025850, &c. of 10, and the index 2, be added; or, which is the same thing, if it be multiplied by its reciprocal 0.434294481903 518, the value of the subtangent of the logarithmic curve, to which Briggs's logarithms are adapted, we shall have the true tabular logarithms of 98, 99, 100, 101, 102. These are to be interpolated by ten intervals, and then we shall have the logarithms of all the numbers between 980 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 $\backslash frac\{\backslash sqrt[10]\{84\; \backslash times\; 1020\}\}\{9945\}\; =\; 2$;

and thus having the logarithms of all the numbers less than 100, you have nothing to do but interpolate the several terms, through ten intervals.

Now the void places may be filled up by the following theorem. Let $n$ be a number, whose logarithm is wanted; let $x$ be the difference between that and the two nearest numbers, equally distant on each side, whose logarithms are already found; and let $d$ be half the difference of their logarithms: then the required logarithm of the number $n$, will be had by adding $d\; +\; \backslash frac\{dx\}\{2n\}\; +\; \backslash frac\{dx^3\}\{12n^3\}$, &c. to the logarithm of the lesser number; for if the numbers are represented by $Cp$, $CG$, $CP$, (ibid. n^o 2.) and the ordinates $p$, $PQ$, be raised; if $n$ be wrote for

$CG$, and $x$ for $GP$, or $Gp$, the area $p\_1QP$, or $\backslash frac\{2x\}\{n\}\; +\; \backslash frac\{x^3\}\{2n^3\}$ Logarithms

$+\; \backslash frac\{x^3\}\{3n^3\}$, &c. will be to the area $p\_1HG$, 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

The two first terms $d\; +\; \backslash frac\{dx\}\{2n\}$ of this series, being sufficient for the construction of a canon of logarithms, even to 14 places of figures, provided the number, whose logarithm 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 produced 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 logarithms, and by their second and third differences, if necessary, the void places may be supplied more expeditiously; the rule aforegoing 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 intercalation of logarithms, when of three numbers the logarithm of the lesser and of the middle number are given, or of the middle number and the greater; and this although the numbers should not be in arithmetical progression. Also by pursuing the steps of this method, rules may be easily discovered for the construction of artificial 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 delighted with the construction of logarithms, at his first setting out in those 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 those 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 logarithms 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 be given, viz. equal to half the sum of the logarithms.

Now the series $y\; \backslash times\; \backslash frac\{1\}\{4z\}\; +\; \backslash frac\{1\}\{24z^3\}\; +\; \backslash frac\{181\}\{15120z^5\}\; +\; \backslash frac\{25200z^7\}\{13\}$, &c. 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. $\backslash frac\{y\}\{4z\}$, is sufficient for producing the

Logarithms 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 10000, the first term will exhibit the logarithm to 18 places of figures; and so this series is of great use in filling up the chilinds omitted by Mr Briggs. For example, it is required to find the logarithm of 20001; the logarithm

0.00000000542813; and if the logarithm of the geometrical mean, viz. 4.301051709302416 be added to the quotient, the sum will be

4.301051709845230 = 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 proportionals 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, &c. 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, &c. lord Napier's logarithm of 10, is to 1.000000000 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, &c. or multiplied by 4342944, &c. the ratio of 1.0000, &c. to 2.30258, &c. as is found by dividing 1.0000, &c. by 2.30258, &c. the quotient in the former, and the product in the latter, will give the correspondent logarithm 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 = 86858963806.

It is evident, from what has been said of the con-

struction 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.

And in division, subtract the logarithm of the divisor from the logarithm of the dividend, the remainder is the logarithm of the quotient.

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.

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, cubing, &c. of any decimal fraction by logarithms, it must be observed, that the first significant figure of the power be put so many places below the place of units, as the index of its logarithm wants of 10, 100, &c. multiplied by the index of the power.

Divide the logarithm of the number by the index of the power, the quotient is the logarithm of the root sought.

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 de-

Logarithms 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.

The following method, communicated by Mr Thomas Atkinson, Esq. of Ballyshannon, Ireland, is much more expeditious and easy.

In any series of numbers in a geometrical progression, beginning from unity, as in the margin, the series is composed of a set of continued proportionals, of which the member standing nearest to unity is the common ratio or rate of the proportion. If over or under these another series is placed, as in the example, of numbers in an arithmetical progression, beginning with nought, and whose common difference is unity, the members of this series are called indexes; for they serve to show how many successive multiplications have been made with the common rate to produce that member of the geometrical progression over which each of these indexes does severally stand.

This theory may be considered in another light. If the square root of 10 (that is, of the common rate) is found, it is a mean proportional between 1 and 10, and becomes a new common rate for a new set of continued proportionals, as in the margin. And if the half of unity, which in the former case was the additional difference of the arithmetical progression, is made the additional difference of this new series, and noted as in the example, a new combination is formed of two series agreeing with the first in these remarkable properties, viz. If any two members of the geometrical progression are multiplied together, the sum of their corresponding indexes will become the index of their product; and conversely, if any one of them is divided by any other, the difference of their indexes will be found to be the index of the quotient. This theory is indefinite; and repeated extractions may be made with any proposed number of decimals, and bisection made of the corresponding indexes, until one has no more number to work with; and each of the mean proportionals thus found between 1 and 10, will be found a member of every new geometrical progression formed by every smaller root; and consequently all the roots thus found, together with their corresponding indexes, have,

Thus knowing that $0.000025175$, or such like, is the logm. of the last quotient, one may have that of 2, if he will but call to mind the following circumstances.

in cases of multiplication or division, the same connection, as has been just described.

Let those successive roots be found, and noted in the form of a table, and, in another column, let the corresponding indexes found by these directions be regularly noted, each opposite to its own roots. These indexes are commonly known by the denomination of logarithms; the roots themselves may be called natural numbers.

These roots are composed of natural numbers seldom or never wanted; but from them the logarithms of such as are of general use may be thus found.

Suppose 2 the proposed number, one must examine the table of roots; there he will find 3.16, &c. &c. the nearest to 2 of those which are greater; and 1.778, &c. &c. also nearest to it of those which are less. He may make a division at his pleasure, either $\backslash frac\{3.16\}\{2\}$ or $\backslash frac\{2\}\{1.77\}$; yet let the choice fall on what will yield the

smallest quotient, and let the $\backslash frac\{2\}\{1.77\}\; =\; 1.1246$, &c. &c. circumstances of the calculation be noted, as in the margin, for future direction. Here $\backslash frac\{1.154\}\{1.124\}\; =\; 1.02$, &c. &c.

$\backslash frac\{2\}\{1.77\}\; =\; 1.1246$. With this quotient let the table be applied to as before, and 1.1246, &c. will be found to be between 1.154, &c. &c. and 1.074, &c. &c. and division to be made as in the example. In this manner one is to proceed with each successive quotient, till at length he has one in which the number of the initial decimal noughts is equal at least, if not greater than that of the significant figures. Here the work of division may be discontinued; and as it will rarely happen, that one will not have an additional initial nought for every division, the number cannot be great in calculations of a moderate extent. Having at last found a quotient such as was described, and supposing the number of decimals to be 10, one may readily find the logm. of that quotient thus:—Suppose the quotient 1.000057968; he is to look into the table of roots for those noted with 5 initial decimal noughts, and from any one of these and its corresponding logm. state thus:

In every case of division, if he has logarithms of quotient and divisor, he has also that of the dividend, by adding the two first together: if he has the logarithm of the dividend, and that of either the divisor

Logarithms or quotient, he may find that of the other; for he has only to subtract what he knows from the logarithm of the dividend, the remainder is what he wants, and lastly, that in every division he made, he took one number from the table of roots whose logarithm is known, being noted in the table, and which he made use of as his direction either as a dividend or a divisor: From these circumstances, one may, by the help of the logarithm just found, discover the logarithm of that number of the last division, whether it be dividend or divisor, which was the quotient of the preceding division; and thus, tracing his own work backwards by his notes from quotient to quotient, he may ever so few or ever so many, he will come at last by addition and subtraction to the logarithm of the proposed number.

By this method, the logarithm of any number within the compass of the table of roots may be found: if a greater is proposed, suppose 9495, it must be made 9.495, and its logarithm found; then it must be restored to the proposed form, and have a proper index noted before the decimals just found. How to do this is too well known to have occasion to mention it here.

—3.301029995664 the logarithm of the fraction given.
7 the power to which it is to be raised.

—19.107209969648 the logarithm of the answer.

This differs from the like work in whole numbers only in this, that, in multiplying the decimals, one has at last 2 to be carried from them to the whole numbers; this is to be considered as +2, then $-3\; \backslash times\; 7\; =\; -21$, and $-21\; +\; 2\; =\; -19$ to be noted the index of the answer. Extraction of the roots is only the converse of this. Suppose —19.107209969648 given, to find that root whose exponent number is 7. As 7 is the exponent number here, one may in his mind multiply it by 2 for a trial, as in common divi-

Suppose —1.4771212545 given, to extract the root of its 5th power.
—1.8954252109 the logarithm of the root.

For 5, the exponent of the root $\backslash times\; 1$ is greater than the index of the given logarithm, and 4 is the remainder. Then —1 becomes the index of the logarithm

The reason for finding the logarithm of the last quotient by the common proportion is this: He who has made a table of roots, will find, by inspection only, that as initial noughts come into the decimal parts of the roots, the significant figures just immediately following them do assume the form of a geometrical progression, descending, whose common rate or divisor is 2, as is just the case with the whole of the decimals of the corresponding logarithms; and that the number of the significant figures ended with this property is generally equal to that of the initial noughts: so far as this, and no farther, the common proportion will hold between the significant figures of the decimals in the roots and the same number of places in the logarithms; and for this reason it was needful to continue the successive divisions till a quotient was found so circumstanced, that its logarithm could be found by the proportion.

The same gentleman hath also favoured us with the following new method

THE easiest way to explain this, is first to give an example of involving such numbers.

but the product = 14 being less than 19, must be rejected; then he may try it —19.107209969648 with 3, this yields 21 for a product. This 3 must be noted with —3.301029995664 a negative sign for the index of the new logarithm. Then, on comparing 19 with 21, the difference is 2. This 2 must be carried as 20 to the decimals, and one must from that carry on the division of the decimals with 7 for a divisor, as is usually done in other cases.

of the root; and 4 = the overplus, is to be carried as 40 to the decimals; and from that, division is to be made with 5 as a divisor for the rest of the work.