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This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online atwww.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.
Title: Encyclopaedia Britannica, 11th Edition, "Logarithm" to "Lord Advocate"Author: VariousRelease date: March 15, 2013 [eBook #42342]Most recently updated: October 23, 2024Language: EnglishCredits: Produced by Marius Masi, Don Kretz and the OnlineDistributed Proofreading Team at http://www.pgdp.net
Title: Encyclopaedia Britannica, 11th Edition, "Logarithm" to "Lord Advocate"
Author: Various
Author: Various
Release date: March 15, 2013 [eBook #42342]Most recently updated: October 23, 2024
Language: English
Credits: Produced by Marius Masi, Don Kretz and the OnlineDistributed Proofreading Team at http://www.pgdp.net
*** START OF THE PROJECT GUTENBERG EBOOK ENCYCLOPAEDIA BRITANNICA, 11TH EDITION, "LOGARITHM" TO "LORD ADVOCATE" ***
Articles in This Slice
LOGARITHM(from Gr.λόγος, word, ratio, andἀριθμός, 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 ax= 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 = logam.
Properties.—The principal properties of logarithms are given by the equations
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 (1/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 10, 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 10, so that log m denotes log10m; but in analytical formulae it is understood to be e.
The logarithms to base 10 of the first twelve numbers to 7 places of decimals are
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 10, 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 0) the mantissa is still kept positive, so that, for example,
log .25613 =1.4084604, 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 10 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 10 as base. The explanation of this property of the base 10 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 10 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 reality1, 8 denotes2, 10 denotes 0, &c. Logarithms thus increased are frequently referred to for the sake of distinction astabular logarithms, so that the tabular logarithm = the true logarithm + 10.
In tables of logarithms of numbers to base 10 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
logab × logba = 1, logbm = logam × (1/logab).
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/logab, which is called themodulusof 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 10; 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/loge10, which is called the modulus of the common system of logarithms. The numerical value of this modulus is 0.43429 44819 03251 82765 11289 ..., and the value of its reciprocal, loge10 (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,
the numerical value of which is,
2.71828 18284 59045 23536 02874 ....
The logarithmic Function.—The mathematical function log x or logex 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−1x,&c., log x and exwhich 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 equationlog x =∫x1dt, (x > 0).tThis equation defines log x for positive values of x; if x ≤ 0 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 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 (xh− 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 (1 + x), viz.log (1 + x) = x −1⁄2x2+1⁄3x3−1⁄4x4+ ...;the series, however, is convergent for real values of x only when x lies between +1 and −1. Other formulae which are deducible from thisequation are given in the portion of this article relating to the calculation of logarithms.The function log x as x increases from 0 towards ∞ steadily increases from −∞ towards +∞. It has the important property that it tends to infinity with x, but more slowly than any power of x,i.e.that x−mlog x tends to zero as x tends to ∞ for every positive value of m however small.Theexponential 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 0 toward ∞ as y increases from -∞ towards +∞. As y tends towards ∞, exp y tends towards ∞ more rapidly than any power of y.The exponential function possesses the properties(i.)exp (x + y) = exp x × exp y.(ii.)(d/dx) exp x = exp x.(iii.)exp x = 1 + x + x2/2! + x3/3! + ...From (i.) and (ii.) it may be deduced thatexp x = (1 + 1 + 1/2! + 1/3! + ... )x,where the right-hand side denotes the positive xth power of the number 1 + 1 + 1/2! + 1/3! + ... usually denoted by e. It is customary, therefore, to denote the exponential function by exand the resultex= 1 + x + x2/2! + x3/3! ...is known as theexponential theorem.The definitions of the logarithmic and exponential functions may be extended to complex values of x. Thus if x = ξ + iηlog x =∫x1dttwhere 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 havelog (ξ + iη) = log √(ξ2+ η2) + i(α + 2nπ),where α is the numerically least angle whose cosine and sine are ξ/√(ξ2+ η2) and η/√(ξ2+ η2), and n denotes any integer. Thus even when the argument is real log x has an infinite number of values; for putting η = 0 and taking ξ positive, in which case α = 0, we obtain for log ξ the infinite system of values log ξ + 2nπi. 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 equationlog(1 + x) = x −1⁄2x2+1⁄3x3−1⁄4x4+ ...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. Alsoexp (ξ − iη) = eξ(cos η + i sin η).An alternative method of developing the theory of the exponential function is to start from the definitionexp x = 1 + 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.
The logarithmic Function.—The mathematical function log x or logex 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−1x,&c., log x and exwhich 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
This equation defines log x for positive values of x; if x ≤ 0 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 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 (xh− 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 (1 + x), viz.
log (1 + x) = x −1⁄2x2+1⁄3x3−1⁄4x4+ ...;
the series, however, is convergent for real values of x only when x lies between +1 and −1. Other formulae which are deducible from thisequation are given in the portion of this article relating to the calculation of logarithms.
The function log x as x increases from 0 towards ∞ steadily increases from −∞ towards +∞. It has the important property that it tends to infinity with x, but more slowly than any power of x,i.e.that x−mlog x tends to zero as x tends to ∞ for every positive value of m however small.
Theexponential 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 0 toward ∞ as y increases from -∞ towards +∞. As y tends towards ∞, exp y tends towards ∞ more rapidly than any power of y.
The exponential function possesses the properties
From (i.) and (ii.) it may be deduced that
exp x = (1 + 1 + 1/2! + 1/3! + ... )x,
where the right-hand side denotes the positive xth power of the number 1 + 1 + 1/2! + 1/3! + ... usually denoted by e. It is customary, therefore, to denote the exponential function by exand the result
ex= 1 + x + x2/2! + x3/3! ...
is known as theexponential theorem.
The definitions of the logarithmic and exponential functions may be extended to complex values of x. Thus if x = ξ + iη
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 (ξ + iη) = log √(ξ2+ η2) + i(α + 2nπ),
where α is the numerically least angle whose cosine and sine are ξ/√(ξ2+ η2) and η/√(ξ2+ η2), and n denotes any integer. Thus even when the argument is real log x has an infinite number of values; for putting η = 0 and taking ξ positive, in which case α = 0, we obtain for log ξ the infinite system of values log ξ + 2nπi. 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(1 + x) = x −1⁄2x2+1⁄3x3−1⁄4x4+ ...
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 (ξ − iη) = eξ(cos η + i sin η).
An alternative method of developing the theory of the exponential function is to start from the definition
exp x = 1 + 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 of 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’sMirifici Logarithmorum Canonis Descriptio ...(Edinburgh, 1614). The work is a small quarto containing fifty-seven pages of explanatory matter and a table of ninety pages (seeNapier, 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 thedifferentiae, 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 baseewheree= 2.7182818...; the relation between N (a sine) and L its logarithm, as defined in theCanonis Descriptio, being N = 107e−L/(l07), so that (ignoring the factors 107, the effect of which is to render sines and logarithms integral to 7 figures), the base is e−1. Napier’s logarithms decrease as the sines increase. If l 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 betweenland L is expressed by
L = 107loge107− 107l or el= 107e−L/(107)
Napier’s work (which will henceforth in this article be referred to as theDescriptio) 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 theDescriptio, 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 publishedCertaine 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 theDescriptio, 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 theDescriptio. 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 theDescriptiowith enthusiasm. In a letter to Archbishop Usher, dated Gresham House, March 10, 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.1I 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.2He brought with him somecalculations he had made, and suggested to Napier the advantages that would result from the choice of 10 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’sLogarithmorum 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,3and 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 hisArithmetica logarithmica, a folio work containing the logarithms of the numbers from l to 20,000, and from 90,000 to 100,000 (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 100,000 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’sArithmetica logarithmica, the title runningArithmetica 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 titleLogarithmicall Arithmetike ... 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 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, 1000 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.”
“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, 1000 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 101,000 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 astronomy in Gresham College. The title of Gunter’s book, which is very scarce, isCanon 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 theArithmetica logarithmicaof 1628 a table giving log sines, tangents and secants for every minute of the quadrant to 10 places;thesewere obtained by calculating the logarithms of the natural sines, &c. given in theThesaurus mathematicusof 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 titleTrigonometria 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 hisTrigonometria artificialis, giving log sines and tangents to every 10 seconds of the quadrant to 10 places. This work also contains the logarithms of numbers from unity to 20,000 taken from theArithmetica logarithmicaof 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 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 10.
TheDescriptiocontained 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 temerè 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 entitledRabdologiarelating 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 titleMirifici 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 ofexplaining the most difficult matters by a certain and easy method in the fewest possible words.
It is important to notice that in theConstructiologarithms are called artificial numbers; and Robert Napier states that the work was composed several years (aliquot annos) before Napier had invented the name logarithm. TheConstructiotherefore may have been written a good many years previous to the publication of theDescriptioin 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 theDescriptio(1614), the concluding paragraph of which is “Verùm 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, quàm 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 theRabdologia(1617) he wrote “Quorum quidem Logarithmorum speciem aliam multò praestantiorem nunc etiam invenimus, & creandi methodum, unà 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 verò doctissimo viro D. Henrico Briggio Londini publico Geometriae Professori, et amico mihi longè charissimo.”
Briggs in the short preface to hisLogarithmorum 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 theConstructio(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 theRabdologia, and in which the logarithm of unity is 0. 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 amicitiâ quae illi cum Patre meo L. M. intercessit, animo libentissimo in se suscepit; creandi methodo, et usuum explanatione Inventori relictis. Nunc autem ipso ex hâc vitâ evocato, totius negotii onus doctissimi Briggii humeris incumbere, et Sparta haec ornanda illi sorte quadam obtigisse videtur.”
In the address prefixed to theArithmetica logarithmica(1625) Briggs bids the reader not to be surprised that these logarithms are different from those published in theDescriptio:—