LOGARITHM

, from the Greek logos ratio, and ari<*>mos number; q. d. ratio of numbers, or perhaps rather number of ratios; the indices of the ratios of numbers to one another; or a series of numbers in arithmetical proportion, corresponding to as many others in geometrical proportion, in such sort that 0 corresponds to, or is the index of 1, in the geometricals. They have been devised for the ease of large arithmetical calculations.|

Thus, 0, 1, 2, 3, 4, &c, indices or Logarithms, the geometrical progressions, or common numbers. 1, 2, 4, 8, 16, &c, or 20, 21, 22, 23, 24, &c, 1, 3, 9, 27, 81, &c, or 30, 31, 32, 33, 34, &c, 1, 10, 100, 1000, 10000, &c, or 100, 101, 102, 103, 104, &c, Where the same indices, or Logarithms, serve equally for any geometric series; and from which it is evident, that there may be an endless variety of sets of Logarithms to the same common numbers, by varying the 2d term 2, or 3, or 10, &c of the geometric series; as this will change the original series of terms whose indices are the numbers 1, 2, 3, &c; and by interpolation the whole system of numbers may be made to enter the geometrical series, and receive their proportional Logarithms, whether integers or decimals.

Or the Logarithm of any given number, is the index of such a power of some other number, as is equal to the given one. So if N be = rn, then the Logarithm of N is n, which may be either positive or negative, and r any number whatever, according to the different systems of Logarithms. When N is 1, then n is = 0, whatever the value of r is; and consequently the Logarithm of <*> is always 0 in every system of Logarithms. When n is = 1, then N is = r; consequently the root r is always the number whose Logarithm is 1, in every system. When r is = 2.718281828459 &c, the indices are the hyperbolic Logarithms; so that n is always the hyperbolic Logarithm of ―(2.718 &c))n. But in the common Logarithms, r is = 10; so that the common Logarithm of any number, is the index of that power of 10 which is equal to the said number; so the common Logarithm of N = 10n, is n the index of the power of 10; for example, 1000, being the 3d power of 10, has 3 for its Logarithm; and if 50 be = 101.69<*>97, then is 1.69897 the common Logarithm of 50. And hence it follows that this decimal series of terms

1000, 100, 10, 1, .1, .01, .001, or 103, 102, 101, 100, 10-1, 10-2, 10-3, have 3, 2, 1, 0, -1, -2, -3, respectively for the Logarithms of those terms.

The Logarithm of a number contained between any two terms of the first series, is included between the two corresponding terms of the latter; and therefore that Logarithm will consist of the same index, whether positive or negative, as the smaller of those two terms, together with a decimal fraction, which will always be positive. So the number 50 falling between 10 and 100, its Logarithm will fall between 1 and 2, being indeed equal to -1.69897 nearly: also the number .05 falling between the terms .1 and .01, its Logarithm will fall between - 1 and - 2, and is indeed = - 2 + .69897, the index of the less term together with the decimal .69897. The index is also called the Characteristic of the Logarithms, and is always an integer, either positive or negative, or else = 0; and it shews what place is occupied by the first significant figure of the given number, either above or below the place of units, being in the former case + or positive; in the latter - or negative.

When the characteristic of a Logarithm is negative, the sign — is commonly set over it, to distinguish it from the decimal part, which, being the Logarithm found in the tables, is always positive: so - 2 + .69897, or the Logarithm of .05, is written thus ―2.69897. But on some occasions it is convenient to reduce the whole expression to a negative form; which is done by making the characteristic less by 1, and taking the arithmetical complement of the decimal, that is, beginning at the left hand, subtract each figure from 9, except the last significant figure, which is subtracted from 10; so shall the remainders form the Logarithm wholly negative: thus the Logarithm of .05, which is ―2.69897 or - 2 + .69897, is also expressed by - 1.30103, which is all negative. It is also sometimes thought more convenient to express such Logarithms entirely as positive, namely by only joining to the tabular decimal the complement of the index to 10; and in this way the above Logarithm is expressed by 8.69897; which is only increasing the indices in the scale by 10.

The Properties of Logarithms.—From the definition of Logarithms, either as being the indices of a series of geometricals, or as the indices of the powers of the same root, it follows that the multiplication of the numbers will answer to the addition of their Logarithms; the division of numbers, to the subtraction of their Logarithms; the raising of powers, to the multiplying the Logarithm of the root by the index of the power; and the extracting of roots, to the dividing the Logarithm of the given number by the index of the root required to be extracted.

So, 1st, Log. ab or of , Log. 18 or of , .

Secondly, , , , Log. 1/2 or , Log. 1/n or .

Thirdly, ; Log. r1/n or of ; ; log. 21/3 or of ; and . So that any number and its reciprocal have the same Logarithm, but with contrary signs; and the sum of the Logarithms of any number and its reciprocal, or complement, is equal to 0.

History and Construction of Logarithms.—The properties of Logarithms hitherto mentioned, or of arithmetical indices to powers or geometricals, with their various uses and properties, as above-mentioned, are taken notice of by Stifelius, in his Arithmetic; and indeed they were not unknown to the ancients; but they come all far short of the use of Logarithms in| Trigonometry, as first discovered by John Napier, baron of Merchiston in Scotland, and published at Edinburgh in 1614, in his Mirifici Logarithmorum Canonis De<*>riptio; which contained a large canon of Logarithms, with the description and uses of them; but their construction was reserved till the sense of the Learned concerning his invention should be known. This work was translated into English by the celebrated Mr. Edward Wright, and published by his son in 1616. In the year 1619, Robert Napier, son of the inventor of Logarithms, published a new edition of his late father's work, together with the promised Construction of the Logarithms, with other miscellaneous pieces written by his father and Mr. Briggs. And in the same year, 1619, Mr John Speidell published his New Logarithms, being an improved form of Napier's.

All these tables were of the kind that have since been called hyperbolical, because the numbers express the areas between the asymptote and curve of the hyperbola. And Logarithms of this kind were also soon after published by several other persons; as by Ursinus in 1619, Kepler in 1624, and some others.

On the first publication of Napier's Logarithms, Henry Briggs, then professor of Geometry in Gresham College in London, immediately applied himself to the study and improvement of them, and soon published the Logarithms of the first 1000 numbers, but on a new scale, which he had invented, viz, in which the Logarithm of the ratio of 10 to 1 is 1, the Logarithm of the same ratio in Napier's system being 2.30258 &c; and in 1624, Briggs published his Arithmetica Logarithmica, containing the Logarithms of 30,000 natural numbers, to 14 places of figures besides the index, in a form which Napier and he had agreed upon together, which is the present form of Logarithms; also in 1633 was published, to the same extent of figures, his Trigonometria Britannica, containing the natural and logarithmic sines, tangents, &c.

With various and gradual improvements, Logarithms were also published successively, by Gunter in 1620, Wingate in 1624, Henrion in 1626, Miller and Norwood in 1631, Cavalerius in 1632 and 1643, Vlacq and Rowe in 1633, Frobenius in 1634, Newton in 1658, Caramuel in 1670, Sherwin in 1706, Gardiner in 1742, and Dodson's Antilogarithmic Canon in the same year; besides many others of lesser note; not to mention the accurate and comprehensive tables in the Tables Portative, and in my own Logarithms lately published, where a complete history of this science may be seen, with the various ways of constructing them that have been invented by different authors.

In Napier's construction of Logarithms, the natural numbers, and their Logarithms, as he sometimes called them, or at other times the artisicial numbers, are supposed to arise, or to be generated, by the motions of points, describing two lines, of which the one is the natural number, and the other its Logarithm, or artificial. Thus, he conceived the line or length of the radius to be described, or run over, by a point moving along it in such a manner, that in equal portions of time it generated, or cut off, parts in a decreasing geometrical progression, leaving the several remainders, or sines, in geometrical progression also; whilst another point described equal parts of an indefinite line, in the same equal portions of time; so that the respective sums of these, or the whole line generated, were always the arithmeticals or Logarithms of the aforesaid natural sines. In this idea of the generation of the Logarithms and numbers, Napier assumed 0 as the Logarithm of the greatest sine or radius; and next he limited his system, not by assuming a particular value to some assigned number, or part of the radius, but by supposing that the two generating points, which, by their motions along the two lines, described the natural numbers and Logarithms, should have their velocities equal at the beginning of those lines. And this is the reason that, in his table, the natural sines and their Logarithms, at the complete quadrant, have equal differences or increments; and this is also the reason why his scale of Logarithms happens accidentally to agree with what have since been called the hyperbolical Logarithms, which have likewise numeral differences equal to those of their natural numbers at the beginning; except only that these latter increase with the natural numbers, while his on the contrary decrease; the Logarithm of the ratio of 10 to 1 being thé same in both, namely 2.30258509 &c.

Having thus limited his system, Napier proceeds, in the posthumous work of 1619, to explain his construction of the Logarithmic canon. This he effects in various ways, but chiefly by generating, in a very easy manner, a series of proportional numbers, and their arithmeticals or Logarithms; and then finding, by proportion, the Logarithms to the natural sines from those of the natural numbers, among the original proportionals; a particular account of which may be seen in my book of Logarithms above mentioned.

The methods above alluded to, relate to Napier's or the hyperbolical system of Logarithms, and indeed are in a manner peculiar to that sort of them. But in an appendix to the posthumous work, mention is made of other methods, by which the common Logarithms, agreed upon by him and Briggs, may be constructed, and which it appears were written after that agreement. One of these methods is as follows: Having assumed 0 for the Logarithm of 1, and 1000 &c for the Logarithm of 10; this Logarithm of 10, and the successive quotients, are to be divided ten times by 5, by which divisions there will be obtained these other ten Logarithms, namely 2000000000, 400000000, 80000000, 16000000, 3200000, 640000, 128000, 25600, 5120, 1024; then this last Logarithm, and its quotients, being divided ten times by 2, will give these other ten Logarithms,

viz, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1. And the numbers answering to these twenty Logarithms are to be found in this manner, viz, Extract the 5th root of 10 (with ciphers), then the 5th root of that root, and so on for ten continual extractions of the 5th root: so shall these ten roots be the natural numbers belonging to the first ten Logarithms above found, in dividing continually by 5. Next, out of the last 5th root is to be extracted the square root, then the square root of this last root, and so on for ten successive extractions of the square root: so shall these last ten roots be the natural numbers corresponding to the Logarithms or quotients arising from| the last ten divisions by the number 2. And from these twenty Logarithms, 1, 2, 4, 8, &c, and their natural numbers, the author observes that other Logarithms and their numbers may be formed, namely by adding the Logarithms, and multiplying their corresponding numbers. But, besides the immense labour of this method, it is evident that this process would generate rather an antilogarithmic canon, such as Dodson's, than the table of Briggs.

Napier next mentions another method of deriving a few of the primitive numbers and their Logarithms, namely, by taking continually geometrical means, first between 10 and 1, then between 10 and this mean, and again between 10 and the last mean, and so on; and then taking the arithmetical means between their corresponèing Logarithms.

He then lays down various relations between numbers and their Logarithms, such as, that the products and quotients of numbers, answer to the sums and differences of their Logarithms; and that the powers and roots of numbers, answer to the products and quotients of the Logæithms when multiplied or divided by the index of the power or root, &c; as also that, of any two numbers, whose Logarithms are given, if each number be raised to the power denoted by the Logarithm of the other, the two r<*>sults will be equal; thus, if x be the Logarithm of any number X, and y the Logarithm of Y, then is . Napier then adverts to another method of making the Logarithms to a few of the prime integer numbers, which is well adapted to the construction of the common table of Logarithms: this method easily follows from what has been said above, and it depends on this property, that the Logarithm of any number in this scale, is one less than the number of places or figures contained in that power of the given number whose exponent is 10000000000, or the Logarithm of 10, at least as to integer numbers, for they really differ by a fraction, as is shewn by Mr. Briggs in his illustrations of these properties; printed at the end of this Appendix to the Construction of Logarithms.

Kepler gave a construction of Logarithms somewhat varied from Napier's. His work is divided into two parts: In the first, he raises a regular and purely mathematical system of proportions, and the measures of them, demonstrating both the nature and principles of the construction of Logarithms, which he calls the measures of ralios: and in the second part, he applies those principles in the actual construction of his table, which contains only 1000 numbers and their Logarithms. The fundamental principles are briefly these: That at the beginning of the Logarithms, their increments or differences are equal to those of the natural numbers: that the natural numbers may be considered as the decreasing cosines of increasing arcs: and that the secants of those arcs at the beginning have the same differences as the cosines, and therefore the same differences as the Logarithms. Then, since the secants are the reciprocals of the cosines of the same arcs, from the foregoing principles, he establishes the following method of raising the first 100 Logarithms, to the numbers 1000, 999, 998, &c, to 900; viz, in this manner: Divide the radius 1000, increased with seven ciphers, by each of these numbers separate- ly, and the quotients will be the secants of those arcs which have the divisors for their cosines; continuing the division to the 8th figure, as it is in that place only that the arithmetical and geometrical means differ. Then by adding continually the arithmetical means between every two successive secants, the sums will be the the series of Logarithms. Or by adding continually every two secants, the successive sums will be the series of the double Logarithms. He then derives all the other Logarithms from these first 100, by common principles.

Briggs first adverts to the methods mentioned above, in the Appendix to Napier's Construction, which methods were common to both these authors, and had doubtless been jointly agreed upon by them. He first gives an example of computing a Logarithm by the property, that the Logarithm is one less than the number of places or figures contained in that power of the given number whose exponent is the Logarithm of 10 with ciphers. Briggs next treats of the other general method of finding the Logarithms of prime numbers, which he thinks is an easier way than the former, at least when many figures are required. This method consists in taking a great number of continued geometrical means between 1 and the given number whose Logarithm is required; that is, first extracting the square root of the given number, then the root of the first root, the root of the 2d root, the root of the 3d root, and so on, till the last root shall exceed 1 by a very small decimal, greater or less according to the intended number of places to be in the Logarithm sought: then finding the Logarithm of this small number, by easy methods described afterwards, he doubles it as often as he made extractions of the square root, or, which is the same thing, he multiplies it by such power of 2 as is denoted by the said number of extractions, and the result is the required Logarithm of the given number; as is evident from the nature of Logarithms.

But as the extraction of so many roots is a very troublesome operation, our author devises some ingenious contrivances to abridge that labour, chiefly by a proper application of the several orders of the differences of numbers, forming the first instance of what may called the differential method; but for a particular description of these methods, see my Treatise of Logarithms, above quoted, pag. 65 &c.

Mr. James Gregory, in his Vera Circuli Hyperbolæ Quadratura, printed at Padua in 1667, having approximated to the hyperbolic asymptotic spaces by means of a series of inscribed and circumscribed polygons, from thence shews how to compute the Logarithms, which are analogous to the areas of those spaces: and thus the quadrature of the hyperbolic spaces became the same thing as the computation of the Logarithms. He here also lays down various methods to abridge the computation, with the assistance of some properties of numbers themselves, by which the Logarithms of all prime numbers under 1000 may be computed, each by one multiplication, two divisions, and the extraction of the square root. And the same subject is farther pursued in his Exercitationes Geometricæ. In this latter place, he first finds an algebraic expression, in an insinite series, for the Logarithm of (1 + a)/1, and then the like for the| Logarithm of ; and as the one series has all its terms positive, while those of the other are alternately positive and negative, by adding the two together, every 2d term is cancelled, and the double of the other terms gives the Logarithm of the product of and , or the Logarithm of the , that is of the ratio of 1 - a to 1 + a: thus, he finds, first , and , theref. , Which may be accounted Mr. James Gregory's method of making Logarithms.

In 1668, Nicholas Mercator published his Logarithmotechnia, five Methodus Construendi Logarithmos, nova, accurata, & facilis; in which he delivers a new and ingenious method for computing the Logarithms upon principles purely arithmetical; and here, in his modes of thinking and expression, he closely follows the <*>elebrated Kepler, in his writings on the same subject; accounting Logarithms as the measures of ratios, or as the number of ratiunculæ contained in the ratio which any number bears to unity. Purely from these principles, then, the number of the equal ratiunculæ contained in some one ratio, as of 10 to 1, being supposed given, our author shews how the Logarithm, or measure, of any other ratio may be found. But this, however, only by-thebye, as not being the principal method he intends to teach, as his last and best. Having shewn, then, that these Logarithms, or numbers of small ratios, or measures of ratios, may be all properly represented by numbers, and that of 1, or the ratio of equality, the Logarithm or measure being always 0, the Logarithm of 10, or the measure of the ratio of 10 to 1, is most conveniently represented by 1 with any number of ciphers; he then proceeds to shew how the measures of all other ratios may be found from this last supposition: and he explains these principles by some examples in numbers.

In the latter part of the work, Mercator treats of his other method, given by an infinite series of algebraic terms, which are collected in numbers by common addition only. He here squares the hyperbola, and finally finds that the hyperbolic Logarithm of 1 + a, is equal to the insinite series &c; which may be considered as Mercator's quadrature of the hyperbola, or his general expression of an hyperbolic Logarithm, in an insinite series.

And this method was farther improved by Dr. Wallis, in the Philos. Trans. for the year 1668. The celebrated Newton invented also the same series for the quadrature of the hyperbola, and the construction of Logarithms, and that before the same were given by Gregory and Mercator, though unknown to one another, as appears by his letter to Mr. Oldenburg, dated October 24, 1676. The explanation and construction of the Logarithms are also farther pursued in his Fluxions, published in 1736 by Mr. Colson.

Dr. Halley, in the Philos. Trans. for the year 1695, gave a very ingenious essay on the construction of Logarithms, intitled, “A most compendious and facile method for constructing the Logarithms, and exemplified and demonstrated from the nature of numbers, without any regard to the hyperbola, with a speedy method for sinding the number from the given Logarithm.”

Instead of the more ordinary definition of Logarithms, viz, ‘numerorum proportionalium æquidifferentes comites,’ the learned author adopts this other, ‘numeri rationum exponentes,’ as better adapted to the principle on which Logarithms are here constructed, considering them as the number of ratiunculæ contained in the given ratios whose Logarithms are in question. In this way he first arrives at the Logarithmic series before given by Newton and others, and afterwards, by various combinations and sections of the ratios, he derives others, converging still faster than the former. Thus he found the Logarithms of several ratios, as below, viz, when multiplied by the modulus peculiar to the scale of Logarithms, &c, the Log. of 1 to 1 + q, &c, the Log. of 1 to 1 - q, &c, the Log. of a to b, or &c, the same Log. of a to b, or &c, the same Log. of a to b, &c, the Log. of √ab to (1/2)z, &c, the same Log. of √ab to (1/2)z; where a, b, q, are any quantitics, and the values of x, y, z, are thus, viz, .

Dr. Halley also, sirst of any, performed the reverse of the problem, by assigning the number to a given Logarithm; viz, &c, or &c. where l is the Logarithm of the ratio of a the less, to b the greater of any two terms.

Mr. Abraham Sharp of Yorkshire made many calculations and improvements in Logarithms, &c. The most remarkable of these were, his quadrature of the circle to 72 places of figures, and his computation of Logarithms to 61 figures, viz, for all numbers to 100, and for all prime numbers to 1100.

The celebrated Mr. Roger Cotes gave to the world a learned tract on the nature and construction of Logarithms: this was first printed in the Philos. Trans. N° 338, and afterwards with his Harmonia Mensurarum in 1722, under the title Logometria. This tract has justly been complained of, as very obscure and intrieate, and the principle is something between that of Kepler and the method of Fluxions. He invented the terms Modulus and Modular ratio, this being the ratio of &c to 1 or of 1 to | &c; that is the ratio of 2.718281828459 &c to 1, or the ratio of 1 to 0.367879441171 &c; the modulus of any system being the measure or Logarithm of that ratio, which in the hyp. Logarithms is 1, and in Briggs's or the common Logarithms is 0.434294481903 &c.

The learned Dr. Brook Taylor gave another method of computing Logarithms in the Philos. Trans. No. 352, which is founded on these three principles, viz, 1st, That the sum of the Logarithms of any two numbers is the Logarithm of the product of those numbers; 2d, That the Logarithm of 1 is 0, and consequently that the nearer any number is to 1, the nearer will its Logarithm be to 0; 3d, That the product of two numbers or factors, of which the one is greater and the other less than 1, is nearer to 1, than that factor is which is on the same side of 1 with itself; so of the two numbers 2/3 and 4/2, the product <*>/9 is less than 1, but yet nearer to it than 2/3 is, which is also less than 1.— And on these principles he founds an ingenious, though not very obvious, approximation to the Logarithms of given numbers.

In the Philos, Trans. a Mr. John Long gave a method of constructing Logarithms, by means of a small table, something in the manner of one of Briggs's methods for the same purpose.

Also in the Philos. Trans. vol. 61, a tract on the construction of Logarithms is given by the ingenious Mr. William Jones. In this method, all numbers are considered as some certain powers of a constant determined root: thus, any number x is considered as the z power of any root r, or x = rz is taken as a general expression for all numbers in terms of the constant root r and a variable exponent z. Now the index z being the Logarithm of the number x, therefore to find this Logarithm, is the same thing as to find what power of the radix r is equal to the number x.

An elegant tract on Logarithms, as a comment on Dr. Halley's method, was also given by Mr. Jones in his Synopsis Palmariorum Matheseos, published in the year 1706.

In the year 1742, Mr. James Dodson published his Anti-logarithmic Canon, containing all Logarithms under 100,000, and their corresponding natural numbers to eleven places of figures, with all their differences and the proportional parts; the whole arranged in the order contrary to that used in the common tables of numbers and Logarithms, the exact Logarithms being here placed first, and their corresponding nearest numbers in the columns opposite to them.

And in 1767, Mr. Andrew Reid published an “Essay on Logarithms,” in which he shews the computation of Logarithms from principles depending on the binomial theorem, and on the nature of the exponents of powers, the Logarithms of numbers being here considered as the exponents of the powers of 10. In this way he brings out the usual series for Logarithms, and exemplifies Dr. Halley's construction of them. But for the particulars of this, and the methods given by the other authors, we must refer to the historical preface to my treatise on Logarithms.

Besides the authors above-mentioned, many others have treated on the subject of Logarithms; among the principal of whom are Leibnitz, Euler, Maclaurin, Wolfius, Keill, and professor Simson in an ingenious geo metrical tract on Logarithms, contained in his posthumous works, elegantly printed at Glasgow in the year 1776, at the expence of the learned Earl Stanhope, and by his lordship disposed of in presents among gentlemen most eminent for mathematical learning.

For the description and uses of Logarithms in numeral calculations, with the shortest method of constructing them, see the Historical Introduction to my Logarithms, pa. 124 & seq.

Briggs's or Common Logarithms, are those that have 1 for the Logarithm of 10, or which have 0.4342944819 &c for the modulus; as has been explained above.

Hyperbolic Logarithms, are those that were computed by the inventor Napier, and called also sometimes Natural Logarithms, having 1 for their modulus, or 2.302585092994 &c for the Logarithm of 10. These have since been called Hyperbolical Logarithms, because they are analogous to the areas of a rightangled hyperbola between the asymptotes and the curve. See Logarithms, also Hyperbola and Asymptotic Space.

Logislic Logarithms, are certain Logarithms of sexagesimal numbers or fractions, useful in astronomical calculations. The Logistic Logarithm of any number of seconds, is the difference between the common Logarithm of that number and the Logarithm of 3600, the seconds in 1 degree.

The chief use of the table of Logistic Logarithms, is for the ready computing a proportional part in minutes and seconds, when two terms of the proportion are minutes and seconds, or hours and minutes, or other such sexagesimal numbers. See the Introd. to my Logarithms, pa. 144.

Imaginary Logarithm, a term used in the Log. of imaginary and negative quantities; such as - a, or √- a2 or a √- 1. The fluents of certain imaginary expressions are also Imaginary Logarithms; as of , or of , &c. See Euler Analys. Insin. vol. i. pa. 72, 74.

It is well known that the expression x./x represents the fluxion of the Logarithm of x, and therefore the fluent of x./x is the Logarithm of x; and hence the fluent of is the Imaginary Logarithm of x.

However, when these Imaginary Logarithms occur in the solutions of problems, they may be transformed into circular arcs or sectors; that is, the Imaginary Logarithm, or imaginary hyperbolic sector, becomes a real circular sector. See Bernoulli Oper. tom. i, pa. 400, and pa. 512. Maclaurin's Fluxions, art. 762. Cotes's Harmon. Mens. pa. 45. Walmesley, Anal. des Mes. pa. 63.

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