LOGARITHM (from Gr. Abyos, word, ratio, and apx0µos, number), in mathematics, a word invented by John Napier to denote a particular class of function discovered by him, and which may be defined as follows: if a, x, m are any three quantities satisfying the equation ae = m, then a is called the base, and x is said to be the logarithm of m to the base a. This relation between x, a, m, may be expressed also by the equation x=log. m.
Properties.—The principal properties of logarithms are given by the equations
loge (mn) =loge m+loge it, log.(m/n) =loge m—log. n, log. Mr =1' loge m, log. ;(m = (1/r) log. m,
which may be readily deduced from the definition of a logarithm. 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 (I/r)th of the logarithm of the quantity.
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 ro, 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 logarithm is also a function of frequent occurrence in analysis, being regarded as a known and recognized function like sin x or tan x; but in mathematical investigations the base generally employed is not to, but a certain quantity usually denoted by the letter e, of value 2.71828 18284 ... .
Thus in arithmetical calculations if the base is not expressed it is understood to be to, so that log m denotes logie m; but in analytical formulae it is understood to be e.
The logarithms to base to of the first twelve numbers to 7 places of decimals are
log I =0•0000000 log 5=0.6989700 log 9=0.9542425
log 2 = 0.30I0300 log 6 =0.7781513 log I0 = I.0000000
log 3 =0.4771213 log 7 =0.8450980 log I I = I.0413927
log 4= 0.6020600 log 8 = 0.9030900 log I2 = I.0791812
The meaning of these results is that
I=I00, 2 =I00•3010300, 3 =100.4771213,
10 = 101, 11= 101.0413927 12 = 101.0791312
The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa. When the base is 1o, 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 = I.4084604, log 2561300 = 6.4084604, &c.
In the case of fractional numbers (i.e. numbers in which the integral part is o) the mantissa is still kept positive, so that, for example,
log .25613 =7.4084604, log •0025613 =_3.4084604, &c.
the minus sign being usually written over the characteristic, and not before it, to indicate that the characteristic only, and not the whole expression, is negative; thus
1.4084604 stands for—1+ '4084604.
The fact that when the base is to the mantissa of the logarithm is independent of the position of the decimal point in the number affords the chief reason for the choice of to as base. The explanation of this property of the base to is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of to, and this corresponds to the addition or subtraction of some integer in the case of the logarithm, the mantissa therefore remaining intact. It should
be mentioned that in most tables of trigonometrical functions, the number to 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 2, 10 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 + lo.
In tables of logarithms of numbers to base to 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.
It follows very simply from the definition of a logarithm that
log. b X loge a = I , logo m = log. m X (I /log. b) .
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 1/log.b, 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 ro; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier 1/logeio, which is called the modulus of the common system of logarithms. The numerical value of this modulus is o•43429 44819 03251 82765 11289 . . ., and the value of its reciprocal, loge 10 (by multiplication by which Briggian logarithms may be converted into Napierian logarithms) is 2.30258 50929 94045 68401 79914 . .
The quantity denoted by e is the series,
I+I+ T + I + I +...
I I.2 1.2.3 1.2.3.4
the numerical value of which is,
2.71828 18284 59045 23536 02874 .
The logarithmic Function.—The mathematical function log x or loge x is one of the small group of transcendental functions, consisting only of the circular functions (direct and inverse) sin x, cos x, &c., arc sin x or sin1 x,&c., log x and ee which are universally treated
I in analysis as known functions. The notation log x is generally employed in English and American works, but on the' continent of Europe writers usually denote the function by 1x or lg x. The logarithmic function is most naturally introduced into analysis by the equation
logy= J tt (x>o).
This equation defines log x for positive values of x; if x10 o the formula ceases to have any meaning. Thus log x is the integral function of I/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as algebraical or circular functions. A connexion with the circular functions, however, appears later when the definition of log x is extended to complex values of x.
A relation which is of historical interest connects the logarithmic function with the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy=const., it is evident that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant a and b from their point of intersection, is proportional to log b/a.
The following fundamental properties of log x are readily deducible from the definition
(i.) log xy=log x+log y.
(ii.) Limit of (x" 1)/h = log x, wken h is indefinitely diminished. Either of these properties might be taken as itself the definition of log x.
There is no series for log x proceeding either by ascending or descending powers of x, but there is an expansion for log (1+x), viz.
log (I+x)=x—zx2+1x3—zx'+ ... ;
the series, however, is convergent for real values of x only when x lies between +1 and  I. Other formulae which are deducible from this
equation are given in the portion of this article relating to the calculation of logarithms.
The function log x as x increases from o towards so steadily increases frokri so towards +so. It has the important property that it tends to Infinity with x, but more slowly than any power of x, i.e. that x" log x tends to zero as x tends to so for every positive value of m however small.
The exponential function, exp x, may be defined as the inverse of the logarithm: thus x=exp y if y = log x. It is positive for all values of y and increases steadily from o toward so as y increases from eo towards + so. As y tends towards so , exp y tends towards so more rapidly than any power of y.
The exponential function possesses the properties
(i.) exp (x+y) =exp x X exp y.
d
(u.) Tx exp x =exp x.
(iii.) exp x = t +x+x2/2 ! + x3/3 ! + .. . From (i.) and (ii.) it may be deduced that
exp x=(1+1+1/2 1 +1/3 ! + ...)r,
where the righthand side denotes the positive xth power of the number 1+I+1/2 ! +1/3 ! + ... usually denoted by e. It is customary, therefore, to denote the exponential function by ea, and the result
ex = I +x+x2/2 ! +x3/3 ! . • •
is known as the exponential theorem.
The definitions of the logarithmic and exponential functions may be extended to complex values of x. Thus if x=E+in,
log x= ( —
Ji t
where the path of integration in the plane of the complex variable t is any curve which does not pass through the origin; but now log x is not a uniform function, that is to say, if x describes a closed curve it does not follow that log x also describes a closed curve: in fact we have
log (E+in) =logs/ (E2+n2)+i(a+2nir),
where a is the numerically least angle whose cosine and sine are
(E'+n2) and n/sl (E'+n2), and n denotes any integer. Thus even when the argument is real log x has an infinite number of values; for putting n=o and taking E positive, in which case a=o, we obtain for log the infinite system of values log E+2n1ri. It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation i_
log(1 +x) =x—1x2+1 9x'—1x4+
.. .
is true only when the analytical modulus of x is less than unity. The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x. Also
exp (E=ef (cos n+i sin ,t).
An alternative method of developing the theory of the exponential function is to start from the definition
exp x = I .lx+x2/2 ! +x3/3 ! + ... ,
the series on the righthand being convergent for all values of x and therefore defining an analytical function of x which is uniform and regular all over the plane.
Invention and Early History of Logarithms.—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 o1 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 first announcement of the invention was made in Napier's hlirifici Logarilhmorum Canonis Descriptio . . . (Edinburgh, 1614). The work is a small quarto containing fiftyseven pages of explanatory matter and a table of ninety pages (see NAPIER, JoHN). 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. The table gives the logarithms of sines for every minute of seven figures; it is arranged semiquadrantally, 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= 1o7e L/Ip7, so that (ignoring the factors Io7, the effect of which is to render sines and logarithms integral to
7 figures), the base is Napier's logarithms decrease as the sines increase. If 1 denotes the logarithm to base e (that is, the socalled "Napierian " or hyperbolic logarithm) and L denotes, as above, " Napier's " logarithm, the connexion between l and L is expressed by
L = Io, loge Io7 — Io71 or et = I07e—L/103
Napier's work (which will henceforth in this article be referred to as the Descriptio) immediately on its appearance in 1614 attracted the attention of perhaps the two most eminent English mathematicians then living—Edward Wright and Henry Briggs. 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. Both Napier and Wright died soon after the publication of the Descriptio, the date of Wright's death being 1615 and that of Napier 1617, but Briggs lived until 1631. Edward Wright, who was a fellow of Caius College, Cambridge, occupies a conspicuous place in the history of navigation. In 1599 he published Certaine errors in Navigation detected and corrected, and he was the author of other works; to him also is chiefly due the invention of the method known as Mercator's sailing. 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. The preface to Wright's edition consists of a translation of the preface to the Descriptio, together with the addition of the following sentences written by Napier himself: " But now some of our countreymen in this Island well affected to these studies, and the more publique good, procured a most learned Mathematician to translate the same into our vulgar English tongue, who after he had finished it, sent the Coppy of it to me, to bee scene and considered on by myself e. I having most willingly and gladly done the same, finde it to bee most exact and precisely conformable to my minde and the originall. Therefore it may please you who are inclined to these studies, to receive it from me and the Translator, with as much good will as we recommend it unto you." There is a short " preface to the reader " by Briggs, and a description of a triangular diagram invented by Wright for finding the proportional parts. The table is printed to one figure less than in the Descriptio. Edward Wright died, as has been mentioned, in 1615, and his son, Samuel Wright, in the preface states that his father " gave much commendation of this work (and often in my hearing) as of very great use to mariners "; and with respect to the translation he says that " shortly after he had it returned out of Scotland, it pleased God to call him away afore he could publish it." The translation was published in 1616. It was also reissued with a new titlepage in 1618.
Henry Briggs, then professor of geometry at Gresham College, London, and afterwards Savilian professor of geometry at Oxford, welcomed the Descriptio with enthusiasm. In a letter to Archbishop Usher, dated Gresham House, March Io, 1615, he wrote, " Napper, lord of Markinston, path 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 accordingly visited Napier in 1615, and stayed with him a whole month.2 He brought with him some
' Dr Thomas Smith thus describes the ardour with which Briggs studied the Descriptio: Hunc in deliciis habuit, in sinu, in manibus, in pectore gestavit, oculisque avidissimis, et mente attentissima, iterum iterumque perlegit, ... " Vitae quorundam eruditissimorum et illustrium virorum (London, 1707).
2 William Lilly's account of the meeting of Napier and Briggs at Merchiston is quoted in the article NAPIER.
calculations he had made, and suggested to Napier the advantages that would result from the choice of ro as a base, an improvement which he had explained in his lectures at Gresham College, and on which he had written to Napier. Napier said that he had already thought of the change, and pointed out a further improvement, viz., that the characteristics of numbers greater than unity should be positive and not negative, as suggested by Briggs. In .1616 Briggs again visited Napier and showed him the work he had accomplished, and, he says, he would gladly have paid him a third visit''in 1617 had Napier's life been spared.
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 l000 to 14 places of decimals. It was published, probably privately, in 1617, after Napier's death,' and there is no author's name, place or date. 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." Briggs's tract of 1617 is extremely rare, and has generally been ignored or incorrectly described. Hutton erroneously states that it contains the logarithms to 8 places, and his account has been followed by most writers. There is a copy in the British Museum.
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 t to 20,000, and from 90,000 to 1oo,000 (and in some copies to 101,000) 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 1oo,000 to 10 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. He designates it, however, only a second edition of Briggs's Arithmetica logarithmica, the title running Arithmetica logarithmica sive Logarithmorum Chiliades centum, . . . editio secunda aucta per Adrianum Vlacq, Goudanum. This table of Vlacq's was published, with an English explanation prefixed, at London in 1631 under the title Logarithmicall Arithmetike .. . London, printed by George Miller, 1631. There are also copies with the titlepage and introduction in French and in Dutch (Gouda, 1628).
Briggs had himself been engaged in filling up the gap, and in a letter to John Pell, written after the publication of Vlacq's work, and dated October 25, 1628, he says:
" My desire was to have those chiliades that are wantinge betwixt 20 and 90 calculated and printed, and I had done them all almost by my selfe, and by some frendes whom my rules had sufficiently informed, and by agreement the busines was conveniently parted amongst us; but I am eased of that charge and care by one Adrian Vlacque, an Hollander, who hathe done all the whole hundred chiliades and printed them in Latin, Dutche and Frenche, moo bookes in these 3 languages, and hathe could them almost all. But he hathe cutt off 4 of my figures throughout; and hathe left out my dedication, and to the reader, and two chapters the 12 and 13, in the rest he hafh not varied from me at all."
The original calculation of the logarithms of numbers from unity to rot,o0o 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. It contains of course many errors, which were gradually discovered and corrected in the course of the next two hundred and fifty years.
The first calculation or publication of Briggian or common iogarithms of trigonometrical functions was made in 162o by Edmund Gunter, who was Briggs's colleague as professor of
' It was certainly published after Napier's death, as Briggs mentions his " librum posthumum." This liber posthumus was the Lonstructio referred to later in this article.astronomy in Gresham College. The title of Gunter's book, which is very scarce, is Canon triangulorum, and it contains logarithmic sines and tangents for every minute of the quadrant to 7 places of decimals. •
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 to places; there were obtained by calculating the logarithms of the natural sines, &c. given in the Thesaurus mathematicus of Pitiscus (1613).
During the last years of his life Briggs devoted himself to the calculation of logarithmic sines, &c. and at the time of his death in 1631 he had all but completed a logarithmic canon to every hundredth of a degree. This work was published by Vlacq at his own expense at Gouda in 1633, under the title Trigonometria Britannica. It contains log sines (to 14 places) and tangents (to to places), besides natural sines, tangents and secants, at intervals of a hundredth of a degree. In the same year Vlacq published at Gouda his Trigonometria artificialis, giving log sines and tangents to every 10 seconds of the quadrant to to places. This work also contains the logarithms of numbers from unity to 20,000 taken from the Arithmetica logarithmica of 1628. Briggs appreciated clearly the advantages of a centesimal, division of the quadrant, and by dividing the degree into hundredth parts instead of into minutes, made a step towards a reformation in this respect, and but for the appearance of Vlacq's work the decimal division of the degree might have become recognized, as is now the case with the corresponding division of the second. 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—Arithmetica logarilhmica (Briggs), 1624; Arithmetica logarithmica (Vlacq), 1628; Trigonometria Britannica (Briggs), 1633; Trigonometric 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 to.
The Descriptio contained only an explanation of the use of the logarithms without any account of the manner in which the canon was constructed. In an " Admonitio " 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." He awaits therefore the judgment and censure of the learned " priusquam caetera in lucem temere prolata lividorum detrectationi exponantur "; and in an " Admonitio " on the last page of the book he states that he will publish the mode of construction of the canon " si huius inventi usum eruditis gratum fore intellexero." Napier, however, did not live to keep this promise. In 1617 he published a small work entitled Rabdologia relating to mechanical methods of performing multiplications and divisions, and in the same year he died.
The proposed work was published in 1619 by Robert Napier, his second son by his second marriage, under the title Mirifici logarithmorum canonis constructio. . . . It consists of two pages of preface followed by sixtyseven pages of text. In the preface Robert Napier says that he has been assured from undoubted authority that the new invention is much thought of by the ablest mathematicians, and that nothing would delight them more than the publication of the mode of construction of the canon. He therefore issues the work to satisfy their desires, although, he states, it is manifest that it would have seen the light in a far more perfect state if his father could have put the finishing touches to it; and he mentions that, in the opinion of the best judges, his father possessed, among other most excellent gifts, in the highest degree the power of
explaining the most difficult matters by a certain and easy method in the fewest possible words.
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. The Constructio therefore may have been written a good many years previous to the publication of the Descriptio in 1614.
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 10, 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. Nihil in ortu perfectum." In some copies, however, this " Admonitio " is absent. In Wright's translation of 1616 Napier has added the sentence—" But because the addition and subtraction of these former numbers may seeme somewhat painfull, I intend (if it shall please God) in a second Edition, to set out such Logarithmes as shall make those numbers above written to fall upon decimal numbers, such as 100,000,000, 200,000,000, 300,000,000, &c., which are easie to be added or abated to or from any other number " (p..19); and in the dedication of the Rabdologia (1617) he wrote " Quorum quidem Logarithmorum speciem aliam multo praestantiorem nunc etiam invenimus, & creandi methodum, una cum eorum usu (si Deus longiorem vitae & valetudinis usuram concesserit) evulgare statuimus; ipsam autem novi canonis supputationem, ob infirmam corporis nostri valetudinem, viris in hoc studii genere versatis relinquimus: imprimis vero doctissimo viro D. Henrico Briggio Londini publico Geometriae Professori, et amico mihi longe charissimo."
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 post humum, 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. He also mentions that he has published some remarks upon the propositions in spherical trigonometry and upon the new species of logarithms by Henry Briggs, "qui novi hujus Canonis supputandi laborem gravissimum, pro singulari amicitia quae illi cum Patre meo L. M. intercessit, animo libentissimo in se suscepit; creandi methodo, et usuum explanatione Inventori relictis. Nunc autem ipso ex lac vita evocato, totius negotii onus doctissimi Briggii humeris incumbere, et Sparta haec ornanda illi sorte quadam obtigisse videtur."
In the address prefixed to the Arithmetica logarithmica (1625) Briggs bids the reader not to be surprised that these logarithms are different from those published in the Descriptio :
" Ego enim, cum meis auditoribus Londini, publice in Collegio Greshamensi horum doctrinam explicarem; animadverti multo futurum commodius, si Logarithmus sinus totius servaretur o (ut in Canone mirifico), Logarithmus autem partis decimae ejusdem sinus totius, nempe sinus 5 graduum, 44, M. 21, s., esset l0000000000. atque ea de re scripsi statim ad ipsum authorem, et quamprimum per anni tempos, et vacationem a publico docendi munere licuit, profectus sum Edinburgum; ubi humanissime ab eo acceptus hacsi per integram mensem: Cum autem inter nos de horum mutatione sermo haberetur; ille se idem dudum sensisse, et cupivisse dicebat: veruntamen istos, quos jam paraverat edendos curasse, donee alios, si per negotia et valetudinem liceret, magis commodos confecisset. Istam autem mutationem ita faciendam censebat, ut o esset Logarithmus unitatis, et t0000000000 sinus totius: quod ego longe commodissimum esse non potui non agnoscere. Coepi igitur, ejus hortatu, rejectis illis quos antea paraveram, de horum calculo serio cogitare; et sequenti aestate iterum profectus Edinburgum, horum quos hic exhibeo praecipuos. ostendi, idem etiam tertia aestate libentissime facturus, si Deus ilium nobis tamdiu superstitem esse voluisset."
There is also a reference to the change of the logarithms on the titlepage 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. Briggs pointed out in his lectures at Gresham College that it would be more convenient that o should stand for the logarithm of the whole sine as in the Descriptio, but that the logarithm of the tenth part of the whole sine should be ro,000,000,000. He wrote also to Napier at once; and as soon as he could he went to Edinburgh to visit him, where, as he was most hospitably received by him, he remained for a whole month. When they conversed about the change of system, Napier said that he had perceived and desired the same thing, but that he had published the tables which he had already prepared, so that they might be used until he could construct others more convenient. But he considered that the change ought to be so made that o should be the logarithm of unity and 10,000,000,000 that of the whole sine, which Briggs could not but admit was by far the most convenient of all. Rejecting therefore, those which he had prepared already, Briggs began, at Napier's advice, to consider seriously the question of the calculation of new tables. 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." According to Hutton's view, the words, " it is to be hoped that his posthumous work " . . . which occur in the preface to the Chilias, were a modest hint that the share Briggs had had in changing the logarithms should be mentioned, and that, as no attention was paid to it, he himself gave the account which appears in the Arithmetica of 1624. There seems, however, no ground whatever for supposing that Briggs meant to express anything beyond his hope that the reason for the alteration would be explained in the posthumous work; and in his own account, written seven years after Napier's death and five years after the appearance of the work itself, he shows no injured feeling whatever, but even goes out of his way to explain that he abandoned his own proposed alteration in favour of Napier's, and, rejecting the tables he had already constructed, began to consider the calculation of new ones. The facts, as stated by Napier and Briggs, are in complete accordance, and the friendship existing between them was perfect and unbroken to the last. 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."
Hutton's suggestions are all the more to be regretted as they occur as a history which is the result of a good deal of investigation and which for years was referred to as an authority by many writers. His prejudice against Napier naturally produced retaliation, and Mark Napier in defending his ancestor has fallen into the opposite extreme of attempting to reduce Briggs to the level of a mere computer. 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. It is printed on the back of the last page of the table itself, and so cannot have been torn out from the copies that are without it. As there could have been no reason for omitting it after it had once appeared, we may assume that the copies which do not have it are those which
were first issued. It is probable, therefore, that Briggs's copy contained no reference to the change, and it is even possible that the "Admonitio " may have been added after Briggs had communicated with Napier. As special attention has not been drawn to the fact that some copies have the " Admonitio " and some have not, different writers have assumed that Briggs did or did not know of the promise contained in the " Admonitio " according as it was present or absent in the copies they had themselves referred to, and this has given rise to some confusion. It may also be remarked that the date frequently assigned to Briggs's first visit to Napier is 1616, and not 1615 as stated above, the reason being that Napier was generally supposed to have died in 1618 until Mark Napier showed that the true date was 1617. When the Descriplio was published Briggs was fiftyseven years of age, and the remaining seventeen years of his life were devoted with steady enthusiasm to extend the utility of Napier's great invention.
The only other mathematician besides Napier who grasped the idea on which the use of logarithm depends and applied it to the construction of a table is Justus Byrgius (Jobst Biirgi), whose work Ariihmetische send geometrische ProgressTabulen ... was published at Prague in 162o, six years after the publication of the Descriptio of Napier. This table distinctly involves the principle of logarithms and may be described as a modified table of antilogarithms. It consists of two series of numbers, the one being an arithmetical and the other a geometrical progression: thus
0, 1.0000 0000 I O, I ,000I 0000 20, 1,0002 0001
990, 1,0099 4967
In the arithmetical column the numbers increase by Io, in the geometrical column each number is derived from its predecessor by multiplication by i•000i. Thus the number lox in the arithmetical column corresponds to Ioe (i.000l)s in the geometrical column; the intermediate numbers being obtained by interpolation. If we divide the numbers in the geometrical column by rob the correspondence is between iox and (1•000l)s, and the table then becomes one of antilogarithms, the base being (1.0001)1/10, viz. for example (I•o0o1)i16•990=x•00994967. The table extends to 230270 in the arithmetical column, and it is
shown that 230270.022 corresponds to 9'9999 9999 or 109 in the geometrical column; this last result showing that
(i•000 1)23027.022 = ro. The first contemporary mention of Byrgius's table occurs on page II of the " Praecepta " prefixed to Kepler's Tabulae Radolphinae (1627); his words are: " apices logistici J. Byrgio multis annis ante editionem Neperianam viam praeiverent ad hos ipsissimos logarithmos. Etsi homo cunctator et secretorum suorum custos foetum in partu destituit, non ad usus publicos educavit." Another reference to Byrgius occurs in a work by Benjamin Bramer, the brotherinlaw and pupil of Byrgius, who, writing in 163o, says that the latter constructed
his table twenty years ago or more.'
As regards priority of publication, Napier has the advantage
by six years, and even fully accepting Bramer's statement, there are grounds for believing that Napier's work dates from
a still earlier period.
The power of to, which occurs as a factor in the tables of both
Napier and Byrgius, was rendered necessary by the fact that the decimal point was not yet in use. Omitting this factor in
' Frisch's Kepleri opera omnia, ii. 834. Frisch thinks Bramer possibly relied on Kepler's statement quoted in the text (" Quibus forte confisus Kepleri verbis Benj. Bramer . . . "). See also vol. vii. p. 298.
The claims of Byrgius are discussed in Kastner's Geschichte der Mathematik, ii. 375, and iii. 14; Montucla's Histoire des mathematigues, ii. Io; Delambre's Histoire de l'astronomie moderne, i. 560; de Morgan's article on " Tables " in the English Cyclopaedia ; Mark Napier's Memoirs of John Napier of Merchiston (1834), p. 391, and Cantor's Geschichte der Mathematik, ii. (1892), 662. See also Gieswald, Justus Byrg als Mathematiker and dessen Einleitung in seine Logarithmen (Danzig, 1856).the case of both tables, the connexion between N a number and L its " logarithm " is
N=(e')'' (Napier), L=(1•000l)ioo (Byrgius),
viz. Napier gives logarithms to base e 1, Byrgius gives antilogarithms to base (1•000i)I'a.
There is indirect evidence that Napier was occupied with logarithms as early as 1594, for in a letter to P. Crugerus from Kepler, dated September 9, 1624 (Frisch's Kepler, vi. 47), there occurs' the sentence: " Nihil autem supra Neperianam rationem ease puto: etsi quidem Scotus quidam literis ad Tychonem 1594 scriptis jam spem fecit Canonis illius Mirifici." It is here distinctly stated that some Scotsman in the year 1594, in g. 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. In connexion with Kepler's statement the following story, told by Anthony Wood in the Alhenae Oxonienses, is of some importance:
" It must be now known, that one Dr Craig, a Scotchman .. coming out of Denmark into his own country, called upon Joh. Neper, Baron of Mercheston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus, as 'tis said), to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it than that it was by proportional numbers. Which hint Neper taking, he desired him at his return to call upon him again. Craig, after some weeks had passed, did so, and Neper then showed him a rude draught of what he called Canon mirabilis logarithmorum. Which draught, with some alterations, he printing in 1614, it came forthwith into the hands of our author Briggs, and into those of Will. Oughtred, from whom the relation of this matter came."
This story, though obviously untrue in some respects, gives valuable information by connecting Dr Craig with Napier and Longomontanus, who was Tycho Brahe's assistant. Dr Craig was John Craig, the third son of Thomas Craig, who was one of the colleagues of Sir Archibald Napier, John Napier's father, in the office of justicedepute. Between John Craig and John Napier a friendship sprang up which may have been due to their common taste for mathematics. There are extant three letters from Dr John Craig to Tycho Brahe, which show that he was on the most friendly terms with him. In the first letter, of,which the date is not given, Craig says that Sir William Stuart has safely delivered to him, " about the beginning of last winter," the book which he sent him. Now Mark Napier found in the library of the university of Edinburgh a mathematical work bearing a sentence in Latin which he translates, " To Doctor John Craig of Edinburgh, in Scotland, a most illustrious man, highly gifted with various and excellent learning, professor of medicine, and exceedingly skilled in the mathematics, Tycho Brahe bath sent this gift, and with his own hand written this at Uraniburg, ed November 1588." As Sir William Stuart was sent to Denmark to arrange the preliminaries of King James's marriage, and returned to Edinburgh on the 15th of November 1588, it would seem probable that this was the volume referred to by Craig. It appears from Craig's letter, to which we may therefore assign the date 1589, that, five years before, he had made an attempt to reach Uranienburg, but had been baffled by the storms and rocks of Norway, and that ever since then he had been longing to visit Tycho. Now John Craig was physician to the king, and in 1590 James VI. spent some days at Uranienburg, before returning to Scotland from his matrimonial expedition. It seems not unlikely therefore that Craig may have accompanied the king in his visit to Uranienburg .2 In any case it is certain that Craig was a friend and correspondent of Tycho's, and it is probable that he was the " Scotus quidam."
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.
was the result of many years of labour and thought,' undertaken with this special object, and it would seem that Napier had seen some prospect of success nearly twenty years before the publication of the Descriptio. It is very evident that no mere hint with regard to the use of proportional numbers could have been of any service to him, but it is possible that the news brought by Craig of the difficulties placed in the progress of astronomy by the labour of the calculations may have stimulated him to persevere in his efforts.
The " new invention in Denmark " to which Anthony Wood refers as having given the hint to Napier was probably the method of calculation called prosthaphaeresis (often written in Greek letters 3rpouOa0aipeoes), which had its origin in the solution of spherical triangles? The method consists in the use of the formula
sin a sin b=2 tcos(ab)cos(a+b)l,
by means of which the multiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines; and, as such products occur in the solution of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only. It seems to be due to Wittich of Breslau, who was assistant for a short time to Tycho Brahe; and it was used by them in their calculations in 1582. Wittich in 1584 made known at Cassel the calculation of one case by this prosthaphaeresis; and Justus Byrgius proved it in such a manner that from his proof the extension to the solution of all triangles could be deduced.3 Clavius generalized the method in his treatise De astrolabio (1593) , lib. i. lemma liii. The lemma is enunciated as follows:
" Quaestiones omnes, quae per sinus, tangentes, atque secantes absolvi solent, per solam prosthaphaeresim, id est, per solam additionem, subtractionem, sine laboriosa numerorum multiplicatione divisioneque expedire."
Clavius then refers to a work of Raymarus Ursus Dithmarsus as containing an account of a particular case. The work is probably the Fundamentum astronomicum (1588). Longomontanus, in his Astronomia Danica (1622), gives an account of the method, stating that it is not to be found in the writings of the Arabs or Regiomontanus. As Longomontanus is mentioned in Anthony Wood's anecdote, and as Wittich as well as Longomontanus were assistants of Tycho, we may infer that Wittich's prosthaphaeresis is the method referred to by Wood.
It is evident that Wittich's prosthaphaeresis could not be a good method of practically effecting multiplications unless the quantities to be multiplied were sines, on account of the labour of the interpolations. 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. In fact, if we put xy=¢e(X+Y), X being a function of x only and Y a function of y only, we can show that we must have X=AeQ', y=Be"; and if we put xy=4(X+Y)4(XY), the solutions are ¢(X+Y)=4(x+)2, and x=sin X, y=sin Y, d,(X+Y) =z cos (X+Y). The former solution gives a method known as that of quartersquares; the latter gives the method of prosthaphaeresis.
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
In the Rabdologia (16x7) he speaks of the canon of logarithms as " a me Longo tempore elaboratum."
A careful examination of the history of the method is given by Scheibel in his Einleitung zur mathematischen Bucherkenntniss, Stuck vii. (Breslau, 1775), pp. 1320; and there is also an account in Kastner's Geschichte der Mathematik, i. 566569 (1796) ; in Montucla's Histoire des mathematiques, i. 583585 and 617619; and in Klugel's Worterbuch (1808), article " Prosthaphaeresis."
Besides his connexion with logarithms and improvements in the method of prosthaphaeresis, Byrgius has a share in the invention of decimal fractions. See Cantor, Geschichte, ii. 567. Cantor attributes to him (in the use of his prosthaphaeresis) the first introduction of a subsidiary angle into trigonometry (vol. ii. 59o).to Napier's Descriptio in .order to describe its reception on the continent, and to mention the other logarithmic tables which were published while Briggs was occupied with his calculations.
John Kepler, who has been 2lready quoted in connexion with Craig's visit to Tycho Brahe, received the invention of logarithms almost as enthusiastically as Briggs. His first mention of the subject occurs in a letter to Schikhart dated the 11th of March 1618, in which he writes—" Extitit Scotus Baro, cujus nomen mihi excidit, qui praeclari quid praestitit, necessitate omni multiplicationum et divisionum in meras additiones et subtractiones commutata, nec sinibus utitur: at tamen opus est ipsi tangentium canone: et varietas, crebritas, difficultasque additionum subtractionumque alicubi laborem multiplicandi et dividendi superat." 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. He says that, although Napier's book had been published five years, he first saw it at Prague two years before; he was then unable to read it, but last year he had met with a little work by Benjamin Ursinus 4 containing the substance of the method, and he at once recognized the importance' of what had been effected. He then explains how he verified the canon, and so found that there were no essential errors in it, although there were a few inaccuracies near the beginning of the quadrant, and he proceeds, " Haec to obiter scire volui, ut quibus to methodis incesseris, quas non dubito et plurimas et ingeniosissimas tibi in promptu esse, eas publici juris fieri, mihi saltem (puto et caeteris) scires fore gratissimum; eoque percepto, tua promissa folio 57, in debitum cecidisse intelligeres." This letter was written two years after Napier's death (of which Kepler was unaware), and in the same year as that in which the Constructio was published. In the same year (1620) Napier's Descriptio (1614) and Constructio (1619) were reprinted by Bartholomew Vincent at Lyons and issued together.,
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 interval of the arguments is so", and the results are given to 8 places; in Napier's canon the interval is 1', and the number of places is 7. The logarithms are strictly Napierian, and the arrangement is identical with that in the canon of 1614. This is the largest Napierian canon that has ever been published.
In the same year (1624) Kepler published at Marburg a table of Napierian logarithms of sines with certain additional coh.;mns to facilitate special calculations.
The first publication of Briggian logarithms on the continent is due to Wingate, who published at Paris in 1625 his Arithmetique logarithmetique, containing sevenfigure logarithms of.
+ The title of this work is—Benjaminis Ursini . . . cursus mathemalici practici volumen primum continens illustr. generosi Dn. Dn. Johannis Neperi Baronet Merchistonij &c. Scott trigonometriam logarithmicam usibus discentium accommodatam . Coloniae .
End of Article: LOGARITHM (from Gr. Abyos, word, ratio, and apx0µos, number) 

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