Bible Encyclopedias
Logarithm

Properties

The principal properties of logarithms are given by the equations log (mn) = log m --Flogs n, loga(m/n) = toga m -logo. n, log. m r = r logo m, logo m= (I /r) logo 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 (r/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 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 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 10, 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 io, so that log m denotes log n m; but in analytical formulae it is understood to be e. 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.

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, 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 -I+ 4084604.

The fact that when the base is io 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 io as base. The explanation of this property of the base io is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of 10, 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 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.

It follows very simply from the definition of a logarithm that logo b X logo a, = 1, 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 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.

The numerical value of this modulus is o4342944 81 9 03251 82765 11289.. ., 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 quantity denoted by e is the series, I I I+ I + i.2 +1.2.3+1.2.3.4 + the numerical value of which is, 2.71828 18284 59 0 45 2 353 6 02874. .

The logarithmic Function

The mathematical function log x or log 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 sin-' x,&c., log x and e x which are universally treated 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 lx or lg x. The logarithmic function is most naturally introduced into analysis by the equation log x= x ? t , (x> o).

This equation defines log x for positive values of x; if o the formula ceases to have any meaning. Thus log x is the integral function of 1/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 bla. The following fundamental properties of log x are readily deducible from the definition (i.) log xy= log x-Flog y. (ii.) Limit of (x h -1) /h =log x, when 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 (I +x), viz.

log ('I' 2x 2 + 3 x 3 -4x 4 +.. .; the series, however, is convergent for real values of x only when x lies between +I and -1. 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 w steadily increases from - co towards +co. 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 00 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 as y increases from -co towards + oo. As y tends towards co, exp y tends towards co more rapidly than any power of y. The exponential function possesses the properties (i.) exp (x+y) =exp x X exp y. d x exp x =exp x.

(iii.) exp x = I -f-x+x 2 /2 ! + x 3 /3 ! -{- .. .

From (i.) and (ii.) it may be deduced that exp x= (I ! +1/3 ! + ...)x, where the right-hand side denotes the positive xth power of the number I +I +1/2 ! +1/3 ! + ... usually denoted by e. It is customary, therefore, to denote the exponential function by e x , and the result ex = I +x+x2/2 ! +x 3 13 !.. .

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= log x =1 - 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) = log,/ Q 2 +7)2) t i (a + 2 n 7r), where a is the numerically least angle whose cosine and sine are (2 +n 2) and 71/,i (t 2 +n 2), and n denotes any integer. Thus even when the argument is real log x has an infinite number of values; for putting 71 =o and taking positive, in which case a = o, we obtain for log the infinite system of values log +2n7ri. 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 log(I +x) =x -",, x2 +3x 3 -4x 4 + 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+in) =e (cos 7 7 +i sin 7,).

An alternative method of developing the theory of the exponential function is to start from the definition exp x = I +x+x2/2 ! x3 /3 !

the series on the right-hand 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 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 first announcement of the invention was made in Napier's Mirifici Logarithmorum Canonis Descriptio.. . (Edinburgh, 1614). The work is a small quarto containing fifty-seven 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 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. If 1 denotes the logarithm to base e (that is, the so-called "Napierian " or hyperbolic logarithm) and L denotes, as above, " Napier's " logarithm, the connexion between 1 and L is expressed by L = r o 7 loge 10 7 - 10 7 / or e t = I 07e-L/Ia7 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 1 599 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 seene and considered on by myselfe. 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 title-page 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 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 accordingly visited Napier in 1615, and stayed with him a whole month. 2 He brought with him some 1 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 virorunt (London, 1707).

William Lilly's account of the meeting of Napier and Briggs at Merchiston is quoted in the article NA Pier.

calculations he had made, and suggested to Napier the advantages) that would result from the choice of io 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. Inl,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 1000 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 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. 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 Arithmctike ... London, printed by George Miller, 1631. There are also copies with the title-page and introduction in French and in Dutch (Gouda, 1628).

Briggs had himself been engaged in filling up the gap, and in a lettex 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 sould 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 hath not varied from me at all." 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. 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 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." This liber posthumus was the Constructio 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 ro 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 10 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 ro seconds of the quadrant to ro 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 - 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. 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." 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 sixty-seven 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 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. 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 Ioo,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 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. 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 hac 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 10000000000. atque ea de re scripsi statim ad ipsum authorem, et quamprimum per anni tempus, et vacationem a publico docendi munere licuit, profectus sum Edinburgum; ubi humanissime ab eo acceptus haesi per integrum 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, donec alios, si per negotia et valetudinem liceret, magis commodos confecisset. Istam autem mutationem ita faciendam censebat, ut o esset Logarithmus unitatis, et 10000000000 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, illi 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 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. 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 Io,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 io,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 Descriptio 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 Arithmetische and geometrische Progress-Tabulen ... was published at Prague in 1620, 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 10, I,0001 0000 20, 1,0002 000 I 9 90, 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 i0001. Thus the number lox in the arithmetical column corresponds to 10 8 (1.0001) x in the geometrical column; the intermediate numbers being obtained by interpolation. If we divide the numbers in the geometrical column by lo g the correspondence is between lox and (I000l) x, and the table then becomes one of antilogarithms, the base being (1.0001) 1 / 10, viz. for example (i. 000r)i. ° 90 = Io0994967. The table extends to 2 3 0270 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 (1.0001)23027 022 = 10. The first contemporary mention of Byrgius's table occurs on page 11 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 brother-in-law and pupil of Byrgius, who, writing in 1630, 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 io, 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 1 Frisch's Kepleri opera amnia, 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 mathematiques, ii. 10; 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. 39 2, 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 1)L (Napier), L = (I. 0001)I 1 oN (Byrgius), viz. Napier gives logarithms to base e ', Byrgius gives antilogarithms to base (I.coo')='a.

There is indirect evidence that Napier was occupied with logarithms as early as 3594, for in a letter to P. Criigerus from Kepler, dated September 9, 1624 (Frisch's Kepler, vi. 47), there occurs the sentence: " Nihil autem supra Neperianam rationem esse 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 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. In connexion with Kepler's statement the following story, told by Anthony Wood in the Athenae 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 justice-depute. 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, 2d 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 3589, 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, 1 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 irpooOa4aipeats), which had its origin in the solution of spherical triangles. 2 The method consists in the use of the formula sin a sin b=2 {cos(a-b)-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=4(X+Y), X being a function of x only and Y a function of y only, we can show that we must have X = Ae gz, y =Be y "; and if we put xy= 0(X+Y) - 43(X-Y), the solutions are 4(X+Y)=4(x+y)2, and x= sin X, y= sin Y, 4(X-1-Y) =-2 cos (X+Y). The former solution gives a method known as that of quarter-squares; 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 1 In the Rabdologia (1617) he speaks of the canon of logarithms as " a me longo tempore elaboratum." 2 A careful examination of the history of the method is given by Scheibel in his Einleitung zur mathematischen Bucherkenntniss, Stuck vii. Breslau, 1 775), pp. 13-20; and there is also an account in Kustner's Geschichte der Mathematik, i. 566-569 (1796); in Montucla's Histoire des mathematiques, i. 583-585 and 617-619; and in Klugel's WOrterbuch (1808), article " Prosthaphaeresis." 3 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. 590).

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 already 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 i ith 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.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 interval of the arguments is io", 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 columns 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 seven-figure logarithms of 4 The title of this work is - Benjaminis Ursini ... cursus mathematici practici volumen primum continens illustr. & generosi Dn. Dn. Johannis Neperi Baronis Merchistonij &c. Scoti trigonometriam logarithmicam usibus discentium accommodatam. Coloniae. .. CIO 10 C XIX. At the end, Napier's table is reprinted, but to two figures less. This work forms the earliest publication of logarithms on the continent.

5 The title is Logarithmorum canonis descriptio, seu arithmeticarum supputationum mirabilis abbreviatio. Ejusque usus in utraque trigonometria ut etiam in omni logistica mathematica, amplissimi, facillimi & expeditissimi explicatio. Authore ac inventore Joanne Nepero, Barone Merchistonii, &c. Scoto. Lugduni.. It will be seen that this title is different from that of Napier's work of 1614; many writers have, however, erroneously given it as the title of the latter.

numbers up to 1000, and log sines and tangents from Gunter's Canon (1620). 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.' Vlacq rendered assistance in the publication of this work, and the privilege is made out to him.

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. The calculation of tables of the natural trigonometrical functions may be said to have formed the work of the last half of the 16th century, and the great canon of natural sines for every 10 seconds to 15 places which had been calculated by Rheticus was published by Pitiscus only in 1613, the year before that in which the Descriptio appeared. In the construction of the natural trigonometrical tables Great Britain had taken no part, and it is remarkable that the discovery of the principles and the formation of the tables that were to revolutionize or supersede all the methods of calculation then in use should have been so rapidly effected and developed in a country in which so little attention had been previously devoted to such questions.

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. It may be remarked that the date usually 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; but it was shown by Mark Napier that the true date is 1617.

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.. .," which contains reprints of Napier's Descriptio of 1614, Kepler's writings on logarithms (1624-1625), &c. In 1889 a translation of Napier's Constructio of 1619 was published by Walter Rae Macdonald. Some valuable notes are added by the translator, in one of which he shows the accuracy of the method employed by Napier in his calculations, and explains the origin of a small error which occurs in Napier's table. Appended to the Catalogue is a full and careful bibliography of all Napier's writings, with mention of the public libraries, British and foreign, which possess copies of each. A facsimile reproduction of Bartholomew Vincent's Lyons edition (1620) of the Constructio was issued in 1895 by A. Hermann at Paris (this imprint occurs on page 62 after the word " Finis ").

It now remains to notice briefly a few of the more important events in the history of logarithmic tables subsequent to the original calculations.

Common or Briggian Logarithms of Numbers. - Nathaniel Roe's Tabulae logarithmicae (1633) was the first complete seven-figure 1 In describing the contents of the works referred to, the language and notation of the present day have been adopted, so that for example a table to radius 10,000,000 is described as a table to 7 places, and so on. 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.

table that was published. 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. All the figures of the number are given at the head of the columns, except the last two, which run down the extreme columns-1 to 50 on the left-hand side, and 50 to loo on the right-hand side. The first four figures of the logarithms are printed at the top of the columns. There is thus an advance half way towards the arrangement now universal in seven-figure tables. 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. The work went through several editions during the 18th century, and was at length superseded in 1785 by Hutton's tables, which continued in successive editions to maintain their position for a century.

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.

In 1742 a seven-figure table was published in quarto form by Gardiner, which is celebrated on account of its accuracy and of the elegance of the printing. A French edition, which closely resembles the original, was published at Avignon in 1770.

In 1783 appeared at Paris the first edition of Francois Callet's tables, which correspond to those of Hutton in England. 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. Vega devoted great attention to the detection and correction of the errors in Vlacq's work of 1628. Vega's Thesaurus has been reproduced photographically by the Italian government. Vega also published in 1797, in 2 vols. 8vo, a collection of logarithmic and trigonometrical tables which has passed through many editions, a very useful one volume stereotype edition having been published in 1840 by Masse. The tables in this work may be regarded as to some extent supplementary to those in Callet.

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. Hutton founded on Sherwin and Callet on Gardiner, and the editions of Vega form a separate offshoot from the original tables. 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. By beginning the table at 20,000 instead of at io,000 the differences are halved in magnitude, while the number of them in a page is quartered. In this table multiples of the differences, instead of proportional parts, are given.' John Thomson of Greenock (1782-1855) made an independent calculation of logarithms of numbers up to 120,000 to 12 places of decimals, and his table has been used to verify the errata already found in Vlacq and Briggs by Lefort (see Monthly Not. R.A.S. vol. 34, P. 447). A table of ten-figure logarithms of numbers up to 100,009 was calculated by W. W. Duffield and published in the Report of the U.S. Coast and Geodetic Survey for 1895-1896 as Appendix 12, pp. 395-7 22. The results were compared with Vega's Thesaurus (1794) before publication.

Common or Briggian Logarithms of Trigonometrical Functions

The next great advance on the Trigonometria artificialis took place more than a century and a half afterwards, when Michael Taylor published in 1792 his seven-decimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the Trigonometria to 10 places and then contracted to 7. On account of the great size of this table, and for other reasons, it never 2 The smallest number of entries which are necessary in a table of logarithms in order that the intermediate logarithms may be calculable by proportional parts has been investigated by J. E. A. Steggall in the Proc. Edin. Math. Soc., 1892, 10, p. 35. This number is 1700 in the case of a seven-figure table extending to ioo,000.

Accounts, of Sang's calculations are given in the Trans. Roy. Soc. Edin., 1872, 26, p. 521, and in subsequent papers in the Proceedings of the same society.

came into very general use, Bagay's Nouvelles tables astronomiques (1829), which also contains log sines and tangents to every second, being preferred; this latter work, which for many years was difficult to procure, has been reprinted with the original title-page and date unchanged. The only other logarithmic canon to every second that has been published forms the second volume of Shortrede's Logarithmic Tables (1849). 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. Prony was charged with the direction of the work, and was expressly required " non seulement a composer des tables qui ne laissassent rien a desirer quant a l'exactitude, mais a en faire le monument de calcul le plus vaste et le plus imposant qui eut jamais ete execute ou meme congu." Those engaged upon the work were divided into three sections: the first consisted of five or six mathematicians, including Legendre, who were engaged in the purely analytical work, or the calculation of the fundamental numbers; the second section consisted of seven or eight calculators possessing some mathematical knowledge; and the third comprised seventy or eighty ordinary computers. The work, which was performed wholly in duplicate, and independently by two divisions of computers, occupied two years. As a consequence of the double calculation, there are two manuscripts, one deposited at the Observatory, and the other in the library of the Institute, at Paris. Each of the two manuscripts consists essentially of seventeen large folio volumes, the contents being as follows: Logarithms of numbers up to 200,000.. 8 vols.

Natural sines. I „ Logarithms of the ratios of arcs to sines from 0400000 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). The tables were all calculated to 14 places, with the intention that only 12 should be published, but the twelfth figure is not to be relied upon. The tables have never been published, and are generally known as the Tables du Cadastre, or, in England, as the great French manuscript tables.

A very full account of these tables, with an explanation of the methods of calculation, formulae employed, &c., was published by Lefort in vol. iv. of the Annales de l'observatoire de Paris. The printing of the table of natural sines was once begun, and Lefort states that he has seen six copies, all incomplete, although including the last page. 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.

As the Tables du Cadastre remained unpublished, other tables appeared in which the quadrant was divided centesimally, the most important of these being Hobert and Ideler's Nouvelles tables trigonometriques (1799), and Borda and Delambre's Tables trigonometriques decimates (1800-1801), both of which are seven-figure tables. The latter work, which was much used, being difficult to procure, and greater accuracy being required, the French government in 1891 published an eight-figure centesimal table, for every ten seconds, derived from the Tables du Cadastre. 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. Antilogarithmic tables are few in number, the only other extensive tables of the same kind that have been published occurring in Shortrede's Logarithmic tables already referred to, and in Filipowski's Table of antilogarithms (1849). Both are similar to Dodson's tables, from which they were derived, but they only give numbers to 7 places.

Hyperbolic or Napierian logarithms (i.e. to base e). - The most elaborate table of hyperbolic logarithms that exists is due to Wolfram, a Dutch lieutenant of artillery. 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. The table appeared in Schulze's Neue and erweiterte Sammlung logarithmischer Tafeln (1778), and was reprinted in Vega's Thesaurus (1794), already referred to. 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. The largest hyperbolic table as regards range was published by Zacharias Dase at Vienna in 1850 under the title Tafel der natiirlichen Logarithmen der Zahlen. Hyperbolic antilogarithms are simple exponentials, i.e. the hyperbolic antilogarithm of x is e x . Such tables can scarcely be said to come under the head of logarithmic tables. See Tables, Mathematical: Exponential Functions. 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 first table of this kind appeared in Kepler's work of 1624 which has been already referred to. The object of a table of log (a/x) is to facilitate the working out of proportions in which the third term is a constant quantity a. In most collections of tables of logarithms, and especially those intended for use in connexion with navigation, there occurs a small table of logistic logarithms in which a = 3600"(= I° or I h), the table giving log 3600 log x, and x being expressed in minutes and seconds. It is also common to find tables in which a= 10800"(=3° or 3 h),and x is expressed in degrees (or hours), minutes and seconds. Such tables are generally given to 4 or 5 places. The usual practice in books seems to be to call logarithms logistic when a is 3600", and proportional when a has any other value.

Addition and Subtraction, or Gaussian Logarithms

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. (1802/3); he calculated a table to 14 places, but only a specimen of it which appeared in the Supplement was printed. The first table that was actually published is due to Gauss, and was printed in Zach's ?llonatliche Correspondenz, xxvi. 498 (1812). Corresponding to the argument log x it gives the values of log (I -Fx - 1) and log (1+x).

Dual Logarithms

This term was used by Oliver Byrne in a series of works published between 1860 and 1870. Dual numbers anti logarithms depend upon the expression of a number as a product of 11, i oi, 1.001. or of. 9, .99, 999 In the preceding résumé only those publications have been mentioned which are of historic importance or interest.' For fuller details with respect to some of these works, for an account of tables published in the latter part of the 19th century, and for those which would now be used in actual calculation, reference should be made to the article Tables, Mathematical.

Calculation of Logarithms

The name logarithm is derived from the words X6 7 wv hp426s, the number of the ratios, and the way of regarding a logarithm which justifies the name may be explained as follows. Suppose that the ratio of 10, or any other particular number, to i is compounded of a very great number of equal ratios, as, for example, 1,000,000, then it can be shown that the ratio of 2 to i is very nearly equal to a ratio compounded of 301,030 of these small ratios, or ratiunculae, that the ratio of 3 to I is very nearly equal to a ratio compounded of 477,121 of them, and so on. The small ratio, or ratiuncula, is in fact that of the millionth root of to to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to I will be nearly the same as that of a301'°30 to i, and so on; or, in other words, if a denotes the millionth root of 10, then 2 will be nearly equal to a 301,030, 3 will be nearly equal to a477,1u, and so on.

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. An account of the processes by which Napier constructed his table was given in the Constructio Canonis of 1619. 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. The processes used by Briggs are explained by him in the preface to the Arithmetica Logarithmica (1624). 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. He first formed the table of numbers and their logarithms: Numbers. Logarithms.

10 I each quantity in the left-hand column being the square root of the one above it, and each quantity in the right-hand column being the half In vol. xv. (1875) of the Verhandelingen of the Amsterdam Academy of Sciences, Bierens de Haan has given a list of 553 tables of logarithms. A previous paper of the same kind, containing notices of some of the tables, was published by him in the Verslagen en Mededeelingen of the same academy (Afd. Natuurkunde) deel. iv. (1862), p. 15.

of the one above it. To construct this table Briggs, using about thirty places of decimals, extracted the square root of io fifty-four times, and thus found that the logarithm of 1.00000 00000 00000 12781 91493 20032 35 was 0.00000 00000 00000 05551 11512 31257 82702, and that for numbers of this form (i.e. for numbers beginning with i followed by fifteen ciphers, and then by seventeen or a less number of significant figures) the logarithms were proportional to these significant figures. He then by means of a simple proportion deduced that log (I00000 00000 00000 I)=o00000 00000 00000 0 434 2 944 81 90325 1804, so that, a quantity 1.00000 00000 00000 x (where x consists of not more than seventeen figures) having been obtained by repeated extraction of the square root of a given number, the logarithm of I00000 00000 00000 x could then be found by multiplying x by 00000 00000 00000 04342 To find the logarithm of 2, Briggs raised it to the tenth power, viz. 1024, and extracted the square root of 1.024 forty-seven times, the result being i 00000 00000 00000 16851 60 57 0 53949 77. Multiplying the significant figures by 4342... he obtained the logarithm of this quantity, viz. 0.00000 00000 00000 0 73 18 5593 6 90623 9336, which multiplied by 2 47 gave 0.01029 995 66 39811 95 265277444, the logarithm of 1.024, true to 17 or 18 places. Adding the characteristic 3, and dividing by 10, he found (since 2 is the tenth root of 1024) log 2 = 30102 9995 6 63981 195. Briggs calculated in a similar manner log 6, and thence deduced log 3.

It will be observed that in the first process the value of the modulus is in fact calculated from the formula.

h I Io^ - i - log e 10' the value of la being 1/2 64, and in the second process log i o 2 is in effect calculated from the formula.

247 X login 2= (22" X lo g e IO IO.

Briggs also gave methods of forming the mean proportionals or square roots by differences; and the general method of constructing logarithmic tables by means of differences is due to him.

The following calculation of log 5 is given as an example of the application of a method of mean proportionals. The process consists in taking the geometric mean of numbers above and below 5, the object being to at length arrive at 5000Ooo. 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. The introduction of infinite series into mathematics effected a great change in the modes of calculation and the treatment of the subject. Besides Napier and Briggs, special reference should be made to Kepler (Chilias, 1624) and Mercator (Logarithmotechnia, 1668), whose methods were arithmetical, and to Newton, Gregory, Halley and Cotes, who employed series. A full and valuable account of these methods is given in Hutton's " Construction of Logarithms," which occurs in the introduction to the early editions of his Mathematical Tables, and also forms tract 21 of his Mathematical Tracts (vol. i., 1812). 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.

Since loge(I +x) =x-2x 2 -3x 3 - 4x4+&c., we have, by changing the sign of x, log e (I - x) _ - x - zx 2 - 3x 3 - x 4 - &c.; whence g 1 +x to=2(x+ix'+1x5+&c.), e l - x and, therefore, replacing x by p +q, log e q =2 p +q +3 () 3T ? (-) 5 + &c., in which the series is always convergent, so that the formula affords a method of deducing the logarithm of one number from that of another.

As particular cases we have, by putting ( q= 1, log e p=2 p 1 +3 (-) 3 l5 5 +&c., and by putting q= p+1, loge(p-f-1) - logep =2 2p+1?-s(2p+1)3+5(2p+1)5-i-&c.

the former of these equations gives a convergent series for logep, and the latter a very convergent series by means of which the logarithm of any number may be deduced from the logarithm of the preceding number.

From the formula for log e (p/q) we may deduce the following very convergent series for log e 2, log e 3 and log e 5, viz.: log e 2=2(7P +5Q +3R), log e 3 =2(11P+8Q +5R), log e 5=2(16P+12Q+7R), where P 1 +3 ?1) 3 5 s (31) &C.

Q 49 + 3 (49) 3 + 6 (49)5 +&c.

R 161 + 3 (161) 3+'5 (161)5+&c.

The following still more convenient formulae for the calculation of log e 2, loge3, &c. were given by J. Couch Adams in the Proc. Roy. Soc., 1878, 27, p. 91. If a= logg = - log (1-10)' 81 (1) c = 10g 80 = log 1 +sr, 126 (8) e =10g1 o = log 1 +1000 then log 2=7a-2b+3c, log 3=IIa-3b+5c, log 5=16a-4b +7c, and log 7 =2(39a - IOb+17c - d) or=19a-4b+8c +e, and we have the equation of condition, a-2b+c=d+2e. By means of these formulae Adams calculated the values of log e 2, log e 3, log e s, and loge? to 276 places of decimals, and deduced the value of log e lo and its reciprocal M, the modulus of the Briggian system of logarithms. The value of the modulus found by Adams is Mo = 0-43429 44 81 9 03251 82765 11289 18916 60508 22 943 97 00 5 80366 65661 14453 78316 58646 4920-8870 77 47292 2 4949 33 8 43 17483 18706 106 74 47 6630-3733 64167 92871 58963 90656 92210 64662 81226 58521 27086 56867 03295 9337 0 86965 88266 88331 16360 773849 0514 28443 48666 76864 65860 85135 56148 212 34 87653 43543 43573 25 which is true certainly to 272, and probably to 273, places (Proc. Roy. Soc., 1886, 42, p. 22, where also the values of the other logarithms are given).

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. By means of these tables and of a factor table we may very readily obtain the Briggian logarithm of a number to 61 or a less number of places or of its hyperbolic logarithm to 48 or a less number of places in the following manner. Suppose the hyperbolic logarithm of the prime number 43,867 required. Multiplying by 50, we have 50X43,867=2,193,350, and on looking in Burckhardt's Table des diviseurs for a number near to this which shall have no prime factor greater than 10,009, it appears that 2,193,349 = 23 X 47 X 2029; thus 43,867= Au (23X47X2029 -{-1), and therefore log e 43,867 = log e 23+loge 47+logy 2029 - log e 50 +2,193,349 2 (2,193,349) 2 + (2,193,349)3 &c. The first term of the series in the second line is whence, taking out the logarithms from Wolfram's table, log e 43, 86 7 = Io68891 76079 60568 10191 3661.

The principle of the method is to multiply the given prime (supposed to consist of 4, 5 or 6 figures) by such a factor that the product may be a number within the range of the factor tables, and such that, when it is increased by I or 2, the prime factors may all be within the range of the logarithmic tables. The logarithm is then obtained by use of the formula d l d2 l d3 2 log e (x+d) = log e x-f- - x2+3 x3 - &c., in which of course the object is to render dlx as small as possible. If the logarithm required is Briggian, the value of the series is to be multiplied by M.

If the number is incommensurable or consists of more than seven figures, we can take the first seven figures of it (or multiply and divide the result by any factor, and take the first seven figures of the result) and proceed as before. An application to the hyperbolic logarithm of is given by Burckhardt in the introduction to his Table des diviseurs for the second million.

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. For example, suppose the logarithm of 543839 required to twelve places. Dividing by 10 5 and by 5 the number becomes i 087678, and resolving this number into factors of the form i - irn we find that 543 8 39 = I O 5 X 5 (I - 1 2 8)(i - 06)(I - 163)(i---.173) X(I_.195)(1 - .197)(1 - ilo 9)(I -. I u 3)(1-1122), where 1-. 1 2 8 denotes i-08, i--. 1 4 6 denotes i-0006, &c., and so on. 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,

9.

The resolution of a number into factors of the above form is easily performed. Taking, for example, the number 1.087678, the object is to destroy the significant figure 8 in the second place of decimals; this is effected by multiplying the number by 1-08, that is, by subtracting from the number eight times itself advanced two places, and we thus obtain 100066376. To destroy the first 6 multiply by 1-0006 giving 1000063361744, and multiplying successively by i-00006 and 1-. 000003, we obtain 1.000000 35793 2, and it is clear that these last six significant figures represent without any further work the remaining factors required. In the corresponding antilogarithmic process the number is expressed as a product of factors of the form 1+.i"x.

This method of calculating logarithms by the resolution of numbers into factors of the form i -. i r n is generally known as Weddle's method, having been published by him in The Mathematician for November 1845, and the corresponding method for antilogarithms by means of factors of the form i+(I) r n is known by the name of Hearn, who published it in the same journal for 1847. In 1846 Peter Gray constructed a new table to 12 places, in which the factors were of the form I - (oi) r n, so that n had the values I, 2,. .. 99; and subsequently he constructed a similar table for factors of the form I +(oi) r n. He also devised a method of applying a table of Hearn's form (i.e. of factors of the form I +I r n) to the construction of logarithms, and calculated a table of logarithms of factors of the form i +(ooi) r n to 24 places. 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. Taking as an example the calculation of the Briggian logarithm of the number 43,867, whose hyperbolic logarithm has been calculated above, we multiply it by 3, giving 131,601, and find by Gray's process that the factors of 1.31601 are (I) 1.316 (5) I. (001)4002 I. 000007 (6) I(ooI)5602 1(001) 2 598 (7) i(o00)6412 i(ooi) 3 780 (8) I (o01)7340 Taking the logarithms from Gray's tables we obtain the required logarithm by addition as follows: In Shortrede's Tables there are tables of logarithms and factors of the form i = (oi) r n to 16 places and of the form i & (. 1) r n to 2 places; and in his Tables de Logarithmes d 2 7 Decimales (Paris, 1867 Fedor Thoman gives tables of logarithms of factors of the form I t i r n. In the Messenger of Mathematics, vol. iii. pp. 66-92, 1873, Henry Wace gave a simple and clear account of both the logarithmic and antilogarithmic processes, with tables of both Briggian and hyperbolic logarithms of factors of the form I t I r n to 20 places.

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. A complete bibliography of this method has been given by A. J. Ellis in a paper "on the potential radix as a means of calculating logarithms," printed in the Proceedings of the Royal Society, vol. xxxi., 1881, pp. 401-407, and vol. xxxii., 1881, pp. 377379. Reference should also be made to Hoppe's Tafeln zur dreissigstelligen logarithmischen Rechnung (Leipzig, 1876), which give in a somewhat modified form a table of the hyperbolic logarithm of +Irn.

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. A full account of this method as applied to the calculation of the Tables du Cadastre is given by Lefort in vol. iv. of the Annales de l'Observatoire de Paris. (J. W. L. G.)