Logarithms Sentence Examples

This work contains the first announcement of logarithms to the world, the first table of logarithms and the first use of the name logarithm, which was invented by Napier.

He speaks of the canon of logarithms as "a me longo tempore elaboratum."

The different editions of the Descriptio and Constructio, as well as the reception of logarithms on the continent of Europe, and especially by Kepler, whose admiration of the invention almost equalled that of Briggs, belong to the history of logarithms (q.v.).

Napier's priority in the publication of the logarithms is unquestioned and only one other contemporary mathematician seems to have conceived the idea on which they depend.

The more one considers the condition of science at the time, and the state of the country in which the discovery took place, the more wonderful does the invention of logarithms appear.

When algebra had advanced to the point where exponents were introduced, nothing would be more natural than that their utility as a means of performing multiplications and divisions should be remarked; but it is one of the surprises in the history of science that logarithms were invented as an arithmetical improvement years before their connexion with exponents was known.

As the deed was not destroyed, but is in existence now, it is to be presumed that the terms of it were, riot fulfilled; but the fact that such a contract should have been drawn up by Napier himself affords a singular illustration of the state of society and the kind of events in the midst of which logarithms had their birth.

The Rabdologia attracted more general attention than the logarithms, and as has been mentioned, there were several editions on the Continent.

During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation.

Steinmetz's formula may be tested by taking a series of hysteresis curves between different limits of B,' measuring their areas by a pianimeter, and plotting the logarithms of these divided by 47r as ordinates against logarithms of the corresponding maximum values of B as abscissae.

Merchiston Academy, housed in the old castle of Napier, the inventor of logarithms, is another institution conducted on English public school lines.

When, by practice with logarithms, we become familiar with the correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including fractional numbers), A /5 acquires a definite meaning as the number corresponding to the extremity of a length x, on the logarithmic scale, such that 5 corresponds to the extremity of 2X.

Thus the concrete fact required to enable us to pass arithmetically from the conception of a fractional number to the conception of a surd is the fact of performing calculations by means of logarithms.

We know that log l oN(I+9) = log l oN+log 10 (I+0), and inspection of a table of logarithms shows that, when 0 is small, log 10 (I+B);s approximately equal to X0, where X is a certain constant, whose value is.

A continued product of this kind can, by taking logarithms, be replaced by an infinite series.

Passing over the invention of logarithms by John Napier, and their development by Henry Briggs and others, the next author of moment was an Englishman, Thomas Harriot, whose algebra (Artis analyticae praxis) was published posthumously by Walter Warner in 1631.

His best-known papers, however, deal with prime numbers; in one of these (" Sur les nombres premiers," 1850) he established the existence of limits within which must be comprised the sum of the logarithms of the primes inferior to a given number.

The principal properties of logarithms are given by the equations log (mn) = log m --Flogs n, loga(m/n) = toga m -logo.

Logarithms were originally invented for the sake of abbreviating arithmetical calculations, as by their means the operations of multiplication and division may be replaced by those of addition and subtraction, and the operations of raising to powers and extraction of roots by those of multiplication and division.

For the purpose of thus simplifying the operations of arithmetic, the base is taken to be Io, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated.

The logarithms to base io of the first twelve numbers to 7 places of decimals are log 1 =0.0000000 log 5 log 2 = 0.3010300 log 6 log 3 =0.477 121 3 log 7 log 4 =0.6020600 log 8 The meaning of these results is that The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa.

When the base is to, the logarithms of all numbers in which the digits are the same, no matter where the decimal point may be, have the same mantissa; thus, for example, log 2.5613 =0-4084604, log 25.613 =1.4084604, log 2561300 = 6.4084604, &c.

It should be mentioned that in most tables of trigonometrical functions, the number io is added to all the logarithms in the table in order to avoid the use of negative characteristics, so that the characteristic 9 denotes in reality 1, 8 denotes a, io denotes o, &c. Logarithms thus increased are frequently referred to for the sake of distinction as tabular logarithms, so that the tabular logarithm =the true logarithm -IIo.

In tables of logarithms of numbers to base io the mantissa only is in general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the rule being that, if the number is greater than unity, the characteristic is less by unity than the number of digits in the integral portion of it, and that if the number is less than unity the characteristic is negative, and is greater by unity than the number of ciphers between the decimal point and the first significant figure.

The second of these relations is an important one, as it shows that from a table of logarithms to base a, the corresponding table of logarithms to base b may be deduced by multiplying all the logarithms in the former by the constant multiplier i/logab, which is called the modulus of the system whose base is b with respect to the system whose base is a.

The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is io; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier /loge io, which is called the modulus of the common system of logarithms.

Other formulae which are deducible from this equation are given in the portion of this article relating to the calculation of logarithms.

The table gives the logarithms of sines for every minute of seven figures; it is arranged semi-quadrantally, so that the differentiae, which are the differences of the two logarithms in the same line, are the logarithms of the tangents.

Napier's logarithms are not the logarithms now termed Napierian or hyperbolic, that is to say, logarithms to the base e where e= 2.7182818 ...; the relation between N (a sine) and L its logarithm, as defined in the Canonis Descriptio, being N=10 7 e L/Ip7, so that (ignoring the factors re, the effect of which is to render sines and logarithms integral to 7 figures), the base is C".

The former translated the work into English; the latter was concerned with Napier in the change of the logarithms from those originally invented to decimal or common logarithms, and it is to him that the original calculation of the logarithmic tables now in use is mainly due.

He at once saw the value of logarithms as an aid to navigation, and lost no time in preparing a translation, which he submitted to Napier himself.

I purpose to discourse with him concerning eclipses, for what is there which we may not hope for at his hands," and he also states " that he was wholly taken up and employed about the noble invention of logarithms lately discovered."

Briggs's Logarithmorum chilias prima, which contains the first published table of decimal or common logarithms, is only a small octavo tract of sixteen pages, and gives the logarithms of numbers from unity to 1000 to 14 places of decimals.

The date of publication is, however, fixed as 1617 by a letter from Sir Henry Bourchier to Usher, dated December 6, 1617, containing the passage- " Our kind friend, Mr Briggs, hath lately published a supplement to the most excellent tables of logarithms, which I presume he has sent to you."

Hutton erroneously states that it contains the logarithms to 8 places, and his account has been followed by most writers.

Briggs continued to labour assiduously at the calculation of logarithms, and in 1624 published his Arithmetica logarithmica, a folio work containing the logarithms of the numbers from to 20,000, and from 00,000 to ioo,000 (and in some copies to roi,000) to 14 places of decimals.

The table occupies 300 pages, and there is an introduction of 88 pages relating to the mode of calculation, and the applications of logarithms.

There was thus left a gap between 20,000 and 90,000, which was filled up by Adrian Vlacq (or Ulaccus), who published at Gouda, in Holland, in 1628, a table containing the logarithms of the numbers from unity to 100,000 to ro places of decimals.

Having calculated 70,000 logarithms and copied only 30,000, Vlacq would have been quite entitled to have called his a new work.

The original calculation of the logarithms of numbers from unity to ror,000 was thus performed by Briggs and Vlacq between 1615 and 1628.

Vlacq's table is that from which all the hundreds of tables of logarithms that have subsequently appeared have been derived.

The first calculation or publication of Briggian or common logarithms of trigonometrical functions was made in 1620 by Edmund Gunter, who was Briggs's colleague as professor of 1 It was certainly published after Napier's death, as Briggs mentions his " librum posthumum."

The next publication was due to Vlacq, who appended to his logarithms of numbers in the Arithmetica logarithmica of 1628 a table giving log sines, tangents and secants for every minute of the quadrant to ro places; there were obtained by calculating the logarithms of the natural sines, &c. given in the Thesaurus mathematicus of Pitiscus (1613).

This work also contains the logarithms of numbers from unity to 20,000 taken from the Arithmetica logarithmica of 1628.

The calculation of the logarithms not only of numbers but also of the trigonometrical functions is therefore due to Briggs and Vlacq; and the results contained in their four fundamental works - A rithmetica logarithmica (Briggs), 1624; Arithmetica logarithmica (Vlacq), 1628; Trigonometria Britannica (Briggs), 1633; Trigonometria artificialis (Vlacq), 1633 - have not been superseded by any subsequent calculations.

In the preceding paragraphs an account has been given of the actual announcement of the invention of logarithms and of the calculation of the tables.

It now remains to refer in more detail to the invention itself and to examine the claims of Napier and Briggs to the capital improvement involved in the change from Napier's original logarithms to logarithms to the base ro.

The Descriptio contained only an explanation of the use of the logarithms without any account of the manner in which the canon was constructed.

It is important to notice that in the Constructio logarithms are called artificial numbers; and Robert Napier states that the work was composed several years (aliquot annos) before Napier had invented the name logarithm.

Passing now to the invention of common or decimal logarithms, that is, to the transition from the logarithms originally invented by Napier to logarithms to the base io, the first allusion to a change of system occurs in the "Admonitio " on the last page of the Descriptio (1614), the concluding paragraph of which is " Verum si huius inventi usum eruditis gratum fore intellexero, dabo fortasse brevi (Deo aspirante) rationem ac methodum aut hunc canonem emendandi, aut emendatiorem de novo condendi, ut ita plurium Logistarum diligentia,limatior tandem et accuratior, quam unius opera fieri potuit, in lucem prodeat.

Briggs in the short preface to his Logarithmorum chilias (1617) states that the reason why his logarithms are different from those introduced by Napier " sperandum, ejus librum posthumum, abunde nobis propediem satisfacturum."

The " liber posthumus " was the Constructio (1619), in the preface to which Robert Napier states that he has added an appendix relating to another and more excellent species of logarithms, referred to by the inventor himself in the Rabdologia, and in which the logarithm of unity is o.

There is also a reference to the change of the logarithms on the title-page of the work.

These extracts contain all the original statements made by Napier, Robert Napier and Briggs which have reference to the origin of decimal logarithms. It will be seen that they are all in perfect agreement.

In the following summer he went to Edinburgh and showed Napier the principal portion of the logarithms which he published in 1624.

These probably included the logarithms of the first chiliad which he published in 1617.

It has been thought necessary to give in detail the facts relating to the conversion of the logarithms, as unfortunately Charles Hutton in his history of logarithms, which was prefixed to the early editions of his Mathematical Tables, and was also published as one of his Mathematical Tracts, has charged Napier with want of candour in not telling the world of Briggs's share in the change of system, and he expresses the suspicion that " Napier was desirous that the world should ascribe to him alone the merit of this very useful improvement of the logarithms."

Briggs assisted Robert Napier in the editing of the " posthumous work," the Constructio, and in the account he gives of the alteration of the logarithms in the Arithmetica of 1624 he seems to have been more anxious that justice should be done to Napier than to himself; while on the other hand Napier received Briggs most hospitably and refers to him as " amico mihi longe charissimo."

In connexion with this controversy it should be noticed that the " Admonitio " on the last page of the Descriptio, containing the reference to the new logarithms, does not occur in all the copies.

Napier gives logarithms to base e ', Byrgius gives antilogarithms to base (I.coo')='a.

It is here distinctly stated that some Scotsman in the year 1594, in a letter to Tycho Brahe, gave him some hope of the logarithms; and as Kepler joined Tycho after his expulsion from the island of Huen, and had been so closely associated with him in his work, he would be likely to be correct in any assertion of this kind.

It satisfies the condition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry; and, analytically considered, it is not so different in principle from the logarithmic method.

An account has now been given of Napier's invention and its publication, the transition to decimal logarithms, the calculation of the tables by Briggs, Vlacq and Gunter, as well as of the claims of Byrgius and the method of prosthaphaeresis.

To complete the early history of logarithms it is necessary to return 1 In the Rabdologia (1617) he speaks of the canon of logarithms as " a me longo tempore elaboratum."

John Kepler, who has been already quoted in connexion with Craig's visit to Tycho Brahe, received the invention of logarithms almost as enthusiastically as Briggs.

This erroneous estimate was formed when he had seen the Descriptio but had not read it; and his opinion was very different when he became acquainted with the nature of logarithms. The dedication of his Ephemeris for 1620 consists of a letter to Napier dated the 28th of July 1619, and he there congratulates him warmly on his invention and on the benefit he has conferred upon astronomy generally and upon Kepler's own Rudolphine tables.

In the same year (1620) Napier's Descriptio (1614) and Constructio (1619) were reprinted by Bartholomew Vincent at Lyons and issued together.5 Napier calculated no logarithms of numbers, and, as already stated, the logarithms invented by him were not to base e.

The first logarithms to the base e were published by John Speidell in his New Logarithmes (London, 1619), which contains hYPerbolic log sines, tangents and secants for every minute of the quadrant to 5 places of decimals.

In 1624 Benjamin Ursinus published at Cologne a canon of logarithms exactly similar to Napier's in the Descriptio of 1614, only much enlarged.

The logarithms are strictly Napierian, and the arrangement is identical with that in the canon of 1614.

In the same year (1624) Kepler published at Marburg a table of Napierian logarithms of sines with certain additional columns to facilitate special calculations.

This work forms the earliest publication of logarithms on the continent.

In the following year, 1626, Denis Henrion published at Paris a Traicte des Logarithmes, containing Briggs's logarithms of numbers up to 20,001 to io places, and Gunter's log sines and tangents to 7 places for every minute.

In the same year de Decker also published at Gouda a work entitled Nieuwe Telkonst, inhoudende de Logarithmi voor de Ghetallen beginnende van r tot io,000, which contained logarithms of numbers up to io,000 to io places, taken from Briggs's Arithmetica of 1624, and Gunter's log sines and tangents to 7 places for every minute.'

The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier's Descriptio.

For more detailed information relating to Napier, Briggs and Vlacq, and the invention of logarithms, the reader is referred to the life of Briggs in Ward's Lives of the Professors of Gresham College (London, 1740); Thomas Smith's Vitae quorundam eruditissimorum et illustrium virorum (Vita Henrici Briggii) (London, 1707); Mark Napier's Memoirs of John Napier already referred to, and the same author's Naperi libri qui supersunt (1839); Hutton's History; de Morgan's article already referred to; Delambre's Histoire de l'Astronomie moderne; the report on mathematical tables in the Report of the British Association for 1873; and the Philosophical Magazine for October and December 1872 and May 1873.

In the years1791-1807Francis Maseres published at London, in six volumes quarto " Scriptores Logarithmici, or a collection of several curious tracts on the nature and construction of logarithms, mentioned in Dr Hutton's historical introduction to his new edition of Sherwin's mathematical tables..

Also, although logarithms have been spoken of as to the base e, &c., it is to be noticed that neither Napier nor Briggs, nor any of their successors till long afterwards, had any idea of connecting logarithms with exponents.

It contains seven-figure logarithms of numbers from I to 100,000, with characteristics unseparated from the mantissae, and was formed from Vlacq's table (1628) by leaving out the last three figures.

The first four figures of the logarithms are printed at the top of the columns.

The final step was made by John Newton in his Trigononometria Britannica (1658), a work which is also noticeable as being the only extensive eightfigure table that until recently had been published; it contains logarithms of sines, &c., as well as logarithms of numbers.

In 1705 appeared the original edition of Sherwin's tables, the first of the series of ordinary seven-figure tables of logarithms of numbers and trigonometrical functions such as are in general use now.

In 1717 Abraham Sharp published in his Geometry Improv'd the Briggian logarithms of numbers from 1 to 100, and of primes from 100 to 1100, to 61 places; these were copied into the later editions of Sherwin and other works.

These tables, which form perhaps the most complete and practically useful collection of logarithms for the general computer that has been published, passed through many editions.

In 1794 Vega published his Thesaurus logarithmorum completus, a folio volume containing a reprint of the logarithms of numbers from Vlacq's Arithmetica logarithmica of 1628, and Trigonometria artificialis of 1633.

The logarithms of numbers are arranged as in an ordinary seven-figure table.

In addition to the logarithms reprinted from the Trigonometria, there are given logarithms for every second of the first two degrees, which were the result of an original calculation.

If we consider only the logarithms of numbers, the main line of descent from the original calculation of Briggs and Vlacq is Roe, John Newton, Sherwin, Gardiner; there are then two branches, viz.

Among the most useful and accessible of modern ordinary sevenfigure tables of logarithms of numbers and trigonometrical functions may be mentioned those of Bremiker, Schriin and Bruhns.

For logarithms of numbers only perhaps Babbage's table is the most convenient.'

In 1871 Edward Sang published a seven-figure table of logarithms of numbers from 20,000 to 200,000, the logarithms between 10o,000 and 200,000 being the result of a new calculation.

In 1784 the French government decided that new tables of sines, tangents, &c., and their logarithms, should be calculated in relation to the centesimal division of the quadrant.

I „ Logarithms of the ratios of arcs to sines from 04 00000 to 0 4.05000, and log sines throughout the quadrant 4 „ Logarithms of the ratios of arcs to tangents from 0 4 00000 to 0 4.05000, and log tangents throughout the quadrant 4 The trigonometrical results are given for every hundred-thousandth of the quadrant (to" centesimal or 3" 24 sexagesimal).

Babbage compared his table with the Tables du Cadastre, and Lefort has given in his paper just referred to most important lists of errors in Vlacq's and Briggs's logarithms of numbers which were obtained by comparing the manuscript tables with those contained in the Arithmetica logarithmica of 1624 and of 1628.

Decimal or Briggian Antilogarithms. - In the ordinary tables of logarithms the natural numbers are all integers, while the logarithms tabulated are incommensurable.

In an antilogarithmic table, the logarithms are exact quantities such as 00001, 00002, &c., and the numbers are incommensurable.

The earliest and largest table of this kind that has been constructed is Dodson's Antilogarithmic canon (1742), which gives the numbers to II places, corresponding to the logarithms from 00001 to .99999 at intervals of 00001.

His table gives the logarithms of all numbers up to 2200, and of primes (and also of a great many composite numbers) from 2200 to 10,009, to 48 decimal places.

Six logarithms omitted in Schulze's work, and which Wolfram had been prevented from computing by a serious illness, were published subsequently, and the table as given by Vega is complete.

Logistic or Proportional Logarithms. - The old name for what are now called ratios or fractions are logistic numbers, so that a table of log (a/x) where x is the argument and a a constant is called a table of logistic or proportional logarithms; and since log (a/x) =log a-log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign.

The usual practice in books seems to be to call logarithms logistic when a is 3600", and proportional when a has any other value.

Gaussian logarithms are intended to facilitate the finding of the logarithms of the sum and difference of two numbers whose logarithms are known, the numbers themselves being unknown; and on this account they are frequently called addition and subtraction logarithms. The object of the table is in fact to give log (a =b) by only one entry when log a and log b are given.

The utility of such logarithms was first pointed out by Leonelli in a book entitled Supplement logarithmique, printed at Bordeaux in the year XI.

Dual numbers anti logarithms depend upon the expression of a number as a product of 11, i oi, 1.001.

Napier's original work, the Descriptio Canonis of 1614, contained, not logarithms of numbers, but logarithms of sines, and the relations between the sines and the logarithms were explained by the motions of points in lines, in a manner not unlike that afterwards employed by Newton in the method of fluxions.

These methods apply, however, specially to Napier's own kind of logarithms, and are different from those actually used by Briggs in the construction of the tables in the Arithmetica Logarithmica, although some of the latter are the same in principle as the processes described in an appendix to the Constructio.

His method of finding the logarithms of the small primes, which consists in taking a great number of continued geometric means between unity and the given primes, may be described as follows.

To every geometric mean in the column of numbers there corresponds the arithmetical mean in the column of logarithms. The numbers are denoted by A, B, C, &c., in order to indicate their mode of formation.

Great attention was devoted to the methods of calculating logarithms during the 17th and 18th centuries.

The earlier methods proposed were, like those of Briggs, purely arithmetical, and for a long time logarithms were regarded from the point of view indicated by their name, that is to say, as depending on the theory of compounded ratios.

Many of the early works on logarithms were reprinted in the Scriptores logarithmici of Baron Maseres already, referred to.

In the following account only those formulae and methods will be referred to which would now be used in the calculation of logarithms.

If the logarithms are to be Briggian all the series in the preceding formulae must be multiplied by M, the modulus; thus, log i o(I +x) = M(x-2x2+3x3-4x4+&c.), and so on.

As has been stated, Abraham Sharp's table contains 61-decimal 10 b= log 24 = - log (1-160) d =10g 49 = - log (1-160) 17253 8 35 62 21868 Briggian logarithms of primes up to I ioo, so that the logarithms of all composite numbers whose greatest prime factor does not exceed this number may be found by simple addition; and Wolfram's table gives 48-decimal hyperbolic logarithms of primes up to 10,009.

The best general method of calculating logarithms consists, in its simplest form, in resolving the number whose logarithm is required into factors of the form I - i r n, where n is one of the nine digits, and making use of subsidiary tables of logarithms of factors of this form.

All that is required therefore in order to obtain the logarithm of any number is a table of logarithms, to the required number of places, of n, 9n, 99 n, 999 n, &c., for n= I, 2, 3,

This method of calculating logarithms by the resolution of numbers into factors of the form i -.

This was published in 1876 under the title Tables for the formation of logarithms and antilogarithms to twenty-four or any less number of places, and contains the most complete and useful application of the method, with many improvements in points of detail.

Although the method is usually known by the names of Weddle and Hearn, it is really, in its essential features, due to Briggs, who gave in the Arithmetica logarithmica of 1624 a table of the logarithms of I + i r n up to r =9 to 15 places of decimals.

It was first formally proposed as an independent method, with great improvements, by Robert Flower in The Radix, a new way of making Logarithms, which was published in 1771; and Leonelli, in his Supplement logarithmique (1802-1803), already noticed, referred to Flower and reproduced some of his tables.

The preceding methods are only appropriate for the calculation of isolated logarithms. If a complete table had to be reconstructed, or calculated to more places, it would undoubtedly be most convenient to employ the method of differences.

The first of a series of ephemerides, calculated on these principles, was published by him at Linz in 1617; and in that for 1620, dedicated to Baron Napier, he for the first time employed logarithms. This important invention was eagerly welcomed by him, and its theory formed the subject of a treatise entitled Chilias Logarithmorum, printed in 1624, but circulated in manuscript three years earlier, which largely contributed to bring the new method into general use in Germany.

Appended were tables of logarithms and of refraction, together with Tycho's catalogue of 777 stars, enlarged by Kepler to 1005.

In 1873 Charles Hermite proved that the base of the Napierian logarithms cannot be a root of a rational algebraical equation of any degree.3 To prove the same proposition regarding 7r is to prove that a Euclidean construction for circle-quadrature is impossible.

The volume contains also dissertations on Logarithms and on the Limits of Quantities and Ratios, and a few problems illustrative of the ancient geometrical analysis.

Similarly the continued fraction given by Euler as equivalent to 1(e - 1) (e being the base of Napierian logarithms), viz.

Of these the most important, besides the few already mentioned, are Tables of Logarithms (1826); Comparative View of the Various Institutions for the Assurance of Lives (1826); Decline of Science in England (1830); Ninth Bridgewater Treatise (1837); The Exposition of 1851 (1851).

The formula then becomes I = Ioe kt (2) where e is the base of Napierian logarithms, and k is a constant which is practically the same as j for bodies which do not absorb very rapidly.

The above definitions of logarithms, &c., relate to cases in which n and p are whole numbers, and are generalized later.

We can, however, denote the result of the process by a symbol, and deal with this symbol according to the laws of arithmetic. In this way we arrive at (i) negative numbers, (ii) fractional numbers, (iii) surds, (iv) logarithms (in the ordinary sense of the word).

Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations.

For a further explanation of logarithms, and for an explanation of the treatment of cases in which an antilogarithm is less than I, see Logarithm.

For practical purposes logarithms are usually calculated to base i o, so that log l o 10 = I, log i n I oo = 2, &C.

To find a root other than a square root we can use logarithms, as explained in § 113.

Logarithms.-Multiplication, division, involution and evolution, when the results cannot be exact, are usually most simply performed, at any rate to a first approximation, by means of a table of logarithms. Thus, to find the square root of 2, we have log A /2 = log (21)=1 log 2.

In 1620 he published his Canon triangulorum (see Logarithms).

Gunter's Line, a logarithmic line, usually laid down upon scales, sectors, &c. It is also called the line of lines and the line of numbers, being only the logarithms graduated upon a ruler, which therefore serves to solve problems instrumentally in the same manner as logarithms do arithmetically.

Homework Once the children come home, Dad supervises the homework, using logarithms invented by John Napier of Edinburgh.

In fact GAP supports finite fields with elements represented via discrete logarithms only up to a given size.

Logarithms - log tables - classic oldfashioned conjuring trick for doing hard sums like magic.

The nature of logarithms is explained by reference to the motion of points in a straight line, and the principle upon which they are based is that of the correspondence of a geometrical and an arithmetical series of numbers.

In the preface to the appendix containing the local arithmetic he states that, while devoting all his leisure to the invention of these abbreviations of calculation, and to examining by what methods the toil of calculation might be removed, in addition to the logarithms, rabdologia and promptuary, he had hit upon a certain tabular arithmetic, whereby the more troublesome operations of common arithmetic are performed on an abacus or chess-board, and which may be regarded as an amusement A facsimile of this document is given by Mark Napier in his Memoirs of John Napier (1834), p. 248.

In the dedication of the former he refers to himself as "mihi jam morbis pene confecto," and in the "Admonitio" at the end he speaks of his "infirma valetudo"; while in the latter he says he has been obliged to leave the calculation of the new canon of logarithms to others "ob infirmam corporis nostri valetudinem."

The logarithms introduced by Napier in the Descriptio are not the same as those now in common use, nor even the same as those now called Napierian or hyperbolic logarithms. The change from the original logarithms to common or decimal logarithms was made by both Napier and Briggs, and the first tables of decimal logarithms were calculated by Briggs, who published a small table, extending to 1000, in 1617, and a large work, Arithmetica Logarithmica, 1 containing logarithms of numbers to 30,000 and from 90,000 to Ioo,000, in 1624.

In that article it is mentioned that a Scotsman in 1594 in a letter to Tycho Brahe held out some hope of logarithms; it is likely that the person referred to is John Craig, son of Thomas Craig, who has been mentioned as one of the colleagues of John Napier's father as justice-depute.

These treatises were probably composed before Napier had invented the logarithms or any of the apparatuses described in the Rabdologia; for they contain no allusion to the principle of logarithms, even where we should expect to find such a reference, and the one solitary sentence where the Rabdologia is mentioned ("sive omnium facillime per ossa Rhabdologiae nostrae") was probably added afterwards.

Apart from the interest attaching to these manuscripts as the work of Napier, they possess an independent value as affording evidence of the exact state of his algebraical knowledge at the time when logarithms were invented.

Besides the logarithms and the calculating rods or bones, Napier's name is attached to certain rules and formulae in spherical trigonometry.

In order to adapt this formula to logarithms, we introduce a subsidiary angle p, such that cot p = cot l cos t; we then have cos D = sin 1 cos( - p) I sin p. In the above formulae our earth is assumed to be a sphere, but when calculating and reducing to the sea-level, a base-line, or the side of a primary triangulation, account must be taken of the spheroidal shape of the earth and of the elevation above the sealevel.

For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS; EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY.

The five processes of deduction then reduce to four, which may be described as (i.) subtraction, (ii.) division, (iii.) (a) taking a root, (iii.) (b) taking logarithms. It will be found that these (and particularly the first three) cover practically all the processes legitimately adopted in the elementary theory of the solution of equations; other processes being sometimes liable to introduce roots which do not satisfy the original equation.

Prony (1755-1839) in the formation of the great French tables of logarithms of numbers, sines, and tangents, and natural sines, called the Tables du Cadastre, in which the quadrant was divided centesimally; these tables have never been published (see Logarithms).

It follows from these equations that the logarithm of the product of any number of quantities is equal to the sum of the logarithms of the quantities, that the logarithm of the quotient of two quantities is equal to the logarithm of the numerator diminished by the logarithm of the denominator, that the logarithm of the rth power of a quantity is equal to r times the logarithm of the quantity, and that the logarithm of the rth root of a quantity is equal to (r/r)th of the logarithm of the quantity.

In an " Adm onitio " on the seventh page Napier states that, although in that place the mode of construction should be explained, he proceeds at once to the use of the logarithms, " ut praelibatis prius usu, et rei utilitate, caetera aut magis placeant posthac edenda, aut minus saltem displiceant silentio sepulta."

We may infer therefore that as early as 1594 Napier had communicated to some one, probably John Craig, his hope of being able to effect a simplification in the processes of arithmetic. Everything tends to show that the invention of logarithms 2 See Mark Napier's Memoirs of John Napier of Merchiston (1834), p. 362.

Ellis in a paper "on the potential radix as a means of calculating logarithms," printed in the Proceedings of the Royal Society, vol.

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The table gives the logarithms of sines for every minute to seven figures.

He contributed two memoirs to the Philosophical Transactions, one, "Logometria," which discusses the calculation of logarithms and certain applications of the infinitesimal calculus, the other, a "Description of the great fiery meteor seen on March 6th, 1716."