# Logarithms Sentence Examples

- JOHN NAPIER (1550-1617), Scottish mathematician and inventor of
**logarithms**, was born at Merchiston near Edinburgh in 1550, and was the eighth Napier of Merchiston. - The table gives the
**logarithms**of sines for every minute to seven figures. - 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." - 2 In this treatise (which was written before Napier had invented the name logarithm)
**logarithms**are called "artificial numbers." - 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. - Fortunately, however, Robert Napier had transcribed his father's manuscript De Arte Logistica, and the copy escaped the fate of the originals in the manner explained in the following note, written in the volume containing them by Francis, seventh Lord Napier: "John Napier of Merchiston, inventor of the
**logarithms**, left his manuscripts to his son Robert, who appears to have caused the following pages to have been written out fair from his father's notes, for Mr Briggs, professor of geometry at Oxford. - The Rabdologia attracted more general attention than the
**logarithms**, and as has been mentioned, there were several editions on the Continent. - 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. - 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. - For write (pq) =sï¿½ and take
**logarithms**of both sides of the fundamental relation; we obtain slox +soot' = + (3ly) S20x 2 +2siixy+s02y 2 = E(aix+(3 ly) 2, &C., and siox+SOly - (S 20 x2 + 2s ii x y+ s ooy 2) +... - 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. - (v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theroem (ï¿½ï¿½ 41, 44)ï¿½ (vi.) Surds and Approximate
**Logarithms**. - From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers. - 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. - Negative Indices and
**Logarithms**. - (i.) Applying the general principles of ï¿½ï¿½ 47-49 to indices, we find that we can interpret Xm as being such that X m .Xm =X 0 =I; i.e. - 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. - ï¿½ ï¿½ If we took
**logarithms**to base a, we should have loga(I+B) =logoIOXXO, approximately. - 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. - Kirchhoff's expression is as follow d+47 r rd l dlog e 167x 2 + t), +t log,: t t I (4) In the above formula e is the base of the Napierian
**logarithms**. The first term on the right-hand side of the equation is the expression for the capacity, neglecting the curved edge distribution of electric force, and the other terms take into account, not only the uniform field between the plates, but also the non-uniform field round the edges and beyond the plates. - 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." - 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. - 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**. - ., and the value of its reciprocal, log e io (by multiplication by which Briggian
**logarithms**may be converted into Napierian**logarithms**) is 2.302585092994 0 45 68401 799 1 4 - The invention of
**logarithms**has been accorded to John Napier, baron of Merchiston in Scotland, with a unanimity which is rare with regard to important scientific discoveries: in fact, with the exception 01 the tables of Justus Byrgius, which will be referred to further on, there seems to have been no other mathematician of the time whose mind had conceived the principle on which**logarithms**depend, and no partial anticipations of the discovery are met with in previous writers. - 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". - Napier's
**logarithms**decrease as the sines increase. - 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. - In a letter to Archbishop Usher, dated Gresham House, March lc), 1615, he wrote, " Napper, lord of Markinston, hath set my head and hands a work with his new and admirable
**logarithms**. I hope to see him this summer, if it please God, for I never saw book which pleased me better, or made me more wonder.' - 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.