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 tt<N
 
 John Napier and the Invention 
 of Logarithms, 1614
 
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 John Napier and the Invention 
 of Logarithms, 1614 
 
 A LECTURE 
 
 BY 
 
 E. W. HOBSON, Sc.D., LL.D., F.R.S. 
 
 sadle irian professor of pure mathematics 
 fellow of Christ's college, Cambridge 
 
 Cambridge : 
 
 at the University Press 
 1914
 
 (Eambrtoge: 
 
 PRINTED BY JOHN CLAY, M.A. 
 AT THE UNIVERSITY PRESS
 
 EMS 
 LIB 
 
 Q\ 
 
 4 
 
 JOHN NAPIER AND THE INVENTION 
 OF LOGARITHMS, 1614 
 
 In the present year there will be held a 
 celebration, under the auspices of the Royal 
 Society of Edinburgh, of the tercentenary of 
 one of the great events in the history of Science, 
 the publication of John Napier's " Mirifici 
 Logarithmorum Canonis Descriptio," a work 
 which embodies one of the very greatest scien- 
 tific discoveries that the world has seen. The 
 invention of Logarithms not only marks an 
 advance of the first importance in Mathematical 
 Science, but as providing a great labour-saving 
 instrument for the use of all those who have 
 occasion to carry out extensive numerical calcu- 
 lations it can be compared in importance only 
 with the great Indian invention of our system 
 of numeration.
 
 6 JOHN NAPIER 
 
 It is almost always extremely instructive to 
 study in detail the form in which a great dis- 
 covery or invention was presented by its 
 originator, and to trace in detail the mode in 
 which the fundamental ideas connected with the 
 discovery shaped themselves in his mind, even 
 when, and just because, later developments or 
 simplifications may have so transformed the under- 
 lying principles, and still more the practice, of the 
 invention, that we have become accustomed to 
 look at the matter from a point of view, at least 
 superficially, very different from the original one 
 of the discoverer. The case of logarithms is very 
 far from being an exception to this rule ; accord- 
 ingly I propose to give an account, as concise as 
 may be, of the conception of a logarithm in the 
 mind of Napier, and of the methods by which he 
 actually constructed his table of logarithms. 
 
 In order fully to appreciate the nature of the 
 difficulties of the task accomplished by the genius 
 of John Napier, some effort of imagination is
 
 JOHN NAPIER 7 
 
 required, to be expended in realizing the narrow- 
 ness of the means available in the early part of 
 the seventeenth century for the calculation of the 
 tables, at a time, before the invention of the 
 Differential and Integral Calculus, when calcula- 
 tion by means of infinite series had not yet been 
 invented. Napier's conception of a logarithm 
 involved a perfectly clear apprehension of the 
 nature and consequences of a certain functional 
 relationship, at a time when no general conception 
 of such a relationship had been formulated, or 
 existed in the minds of Mathematicians, and 
 before the intuitional aspect of that relationship 
 had been clarified by means of the great invention 
 of coordinate geometry made later in the century 
 by Ren£ Descartes. A modern Mathematician 
 regards the logarithmic function as the inverse of 
 an exponential function ; and it may seem to us, 
 familiar as we all are with the use of operations 
 involving indices, that the conception of a loga- 
 rithm would present itself in that connection as a
 
 8 JOHN NAPIER 
 
 fairly obvious one. We must however remember 
 that, at the time of Napier, the notion of an index, 
 in its generality, was no part of the stock of ideas 
 of a Mathematician, and that the exponential 
 notation was not yet in use. 
 
 Summary of the life of Napier. 
 
 I must content myself with giving an exceed- 
 ingly brief account of the external facts of the life 
 of Napier*. 
 
 John Napier^, the eighth Napier of Merchiston, 
 usually described as Baron, or Fear, of Merchiston, 
 was born at Merchiston near Edinburgh in 1550, 
 when his father Archibald Napier was little more 
 than sixteen years old. John Napier matriculated 
 at St Andrews in 1563, but did not stay there 
 
 * For a full account of the life and activities of Napier the 
 " Memoirs of John Napier of Merchiston " by Mark Napier, 
 published in 1834, may be consulted. 
 
 t The name Napier was spelled in various ways, several 
 of which were used by John Napier; thus we find Napeir, 
 Nepair, Nepeir, Neper, Nepper, Naper, Napare, Naipper.
 
 JOHN NAPIER 9 
 
 sufficiently long to graduate, as he departed 
 previous to 1566 in order to pursue his studies 
 on the Continent, whence he returned to Mer- 
 chiston in or before 1571. His first marriage, by 
 which he had one son Archibald who was raised 
 to the peerage in 1627 as Lord Napier, and one 
 daughter, took place in 1572. A few years after 
 the death of his wife in 1579, he married again. 
 By his second marriage he had five sons and 
 five daughters ; the second son, Robert, was his 
 literary executor. The invasion of the Spanish 
 Armada in 1588 led Napier, as an ardent Protes- 
 tant, to take a considerable part in Church politics. 
 In January 159^ he published his first work "A 
 plaine discovery of the whole Revelation of 
 St John." This book is regarded as of consider- 
 able importance in the history of Scottish theo- 
 logical literature, as it contained a method of 
 interpretation much in advance of the age ; it 
 passed through several editions in English, French, 
 German, and Dutch.
 
 io JOHN NAPIER 
 
 In July 1594, Napier entered into a curious 
 contract with a turbulent baron, Robert Logan of 
 Restalrig, who had just been outlawed. In this 
 contract, which appears to shew that John Napier 
 was not free from the prevalent belief in Magic, 
 he agreed to endeavour to discover a treasure 
 supposed to lie hidden in Logan's dwelling-place, 
 Fast Castle. Napier was to receive a third part 
 of the treasure when found, in consideration that 
 "the said J hone sail do his utter & exact dili- 
 gens to serche & sik out, and be al craft & ingyne 
 that he dow, to tempt, trye, and find out the 
 sam, and be the grace of God, ather sail find the 
 sam, or than mak it suir that na sik thing hes 
 been thair ; sa far as his utter trawell diligens and 
 ingyne may reach." 
 
 In a document dated June 7, 1596, Napier 
 gave an account of some secret inventions he had 
 made which were "proffitabill & necessary in 
 theis dayes for the defence of this Hand & with- 
 standing of strangers enemies of God's truth &
 
 JOHN NAPIER ii 
 
 relegion." His activities in this direction were no 
 doubt stimulated by the fear of the generally 
 expected invasion by Philip of Spain. It is 
 interesting to note them, in view of the military 
 tastes of many of his descendants. The inventions 
 consisted of a mirror for burning the enemies' 
 ships at any distance, of a piece of artillery capable 
 of destroying everything round an arc of a circle, 
 and of a round metal chariot so constructed that 
 its occupants could move it rapidly and easily, 
 while firing out through small holes in it. Napier's 
 practical bent of mind was also exhibited in the 
 attention he paid to agriculture, especially on the 
 Merchiston estate, where the land was tilled by a 
 system of manuring with salt. 
 
 There is evidence that Mathematics occupied 
 Napier's attention from an early age. From a 
 MS. that was first published in 1839 under the 
 title " De Arte Logistica " it appears that his 
 investigations in Arithmetic and Algebra had led 
 him to a consideration of the imaginary roots of
 
 12 JOHN NAPIER 
 
 equations, and to a general method for the ex- 
 traction of roots of numbers of all degrees. But, 
 led probably by the circumstances of the time, he 
 put aside this work in order to devote himself to 
 the discovery of means of diminishing the labour 
 involved in numerical computations. The second 
 half of the sixteenth century was the time in which 
 the Mathematicians of the Continent devoted a 
 great deal of attention to the calculation of tables 
 of natural trigonometrical functions. The most 
 prominent name in this connection is that of 
 Georg Joachim Rheticus, the great computer 
 whose work has never been superseded, and the 
 final result of whose labours is embodied in the 
 table of natural sines for every ten seconds to 
 fifteen places of decimals, published by Pitiscus in 
 1613, the year before the publication by Napier 
 of the discovery which was destined to revolu- 
 tionize all the methods of computation, and to 
 substitute the use of logarithmic for that of natural 
 trigonometrical functions. It was in the early
 
 JOHN NAPIER 13 
 
 years of the seventeenth century that Johannes 
 Kepler was engaged in the prodigious task of 
 discovering, and verifying by numerical calculation, 
 the laws of the motion of the planets. In this age 
 of numerical calculation then Napier occupied 
 himself with the invention of methods for the ' 
 diminution of the labour therein involved. He 
 himself states in his " Rabdologia," to which 
 reference will presently be made, that the 
 canon of logarithms is "a me longo tempore 
 elaboratum." It appears from a letter of Kepler 
 that a Scotsman, probably Thomas Craig, a 
 friend of the Napier family, gave the astronomer 
 Tycho Brahe in the year 1594 hopes that an 
 important simplification in the processes of arith- 
 metic would become available. There is strong 
 evidence that Napier communicated his hopes to 
 Craig twenty years before the publication of the 
 Canon. 
 
 The " Descriptio," of which an account will 
 be given presently, was as stated at the outset
 
 i 4 JOHN NAPIER 
 
 published in 1614. About the same time Napier 
 devised several mechanical aids for the perform- 
 ance of multiplications and divisions and for the 
 extraction of square and cube roots. He published 
 an account of these inventions in 161 7 in his 
 " Rabdologia," as he says, " for the sake of those 
 who may prefer to work with the natural numbers." 
 The method which Napier calls Rabdologia 
 consists of calculation of multiplications and 
 divisions by means of a set of rods, usually called 
 "Napier's bones." In 1617, immediately after 
 the publication of the " Rabdologia," Napier 
 died. 
 
 The " Descriptio " did not contain an account 
 of the methods by which the " wonderful canon " 
 was constructed. In an "Admonitio" printed at 
 the end of Chapter 11, Napier explains that he 
 prefers, before publishing the method of construc- 
 tion, to await the opinion of the learned world 
 on the canon ; he says " For I expect the judge- 
 ment & censure of learned men hereupon, before
 
 JOHN NAPIER 15 
 
 the rest rashly published, be exposed to the 
 detraction of the envious." 
 
 The " Mirifici Logarithmorum Canonis Con- 
 struction' which contains a full explanation of the 
 method of construction of the wonderful canon, 
 and a clear account of Napier's theory of loga- 
 rithms, was published by his son Robert Napier 
 in 16 1 9. In the preface by Robert Napier it is 
 stated that this work was written by his father 
 several years before the word "logarithm" was 
 invented, and consequently at an earlier date than 
 that of the publication of the "Descriptio." In 
 the latter the word " logarithm " is used through- 
 out, but in the " Constructio," except in the title, 
 logarithms are called " numeri artificiales." After 
 explaining that the author had not put the finish- 
 ing touch to the little treatise, the Editor writes 
 "Nor do I doubt that this posthumous work 
 would have seen the light in a much more perfect 
 & finished state, if God had granted a longer 
 enjoyment of life to the Author, my most dearly
 
 1 6 JOHN NAPIER 
 
 beloved father, in whom, by the opinion of the 
 wisest men, among other illustrious gifts this 
 shewed itself pre-eminent, that the most difficult 
 matters were unravelled by a sure and easy 
 method, as well as in the fewest words." 
 
 Reception of the Canon by Contemporary 
 Mathematicians. 
 
 The new invention attracted the attention 
 ol British and Foreign Mathematicians with a 
 rapidity which may well surprise us when we take 
 into account the circumstances of the time. In 
 particular the publication of the wonderful canon 
 was received by Kepler with marked enthusiasm. 
 In his "Ephemeris" for 1620, Kepler published 
 as the dedication a letter addressed to Napier, 
 dated July 28, 161 9, congratulating him warmly 
 on his invention and on the benefit he had con- 
 ferred upon Astronomy. Kepler explains how he 
 verified the canon and found no essential errors
 
 JOHN NAPIER 17 
 
 in it, beyond a few inaccuracies near the begin- 
 ning of the quadrant. The letter was written two 
 years after Napier's death, of which Kepler had 
 not heard. In 1624 Kepler himself published a 
 table of Napierian logarithms with modifications 
 and additions. The " Descriptio," on its publi- 
 cation in 1 6 14, at once attracted the attention of 
 Henry Briggs (1 556-1 630), Fellow of St John's 
 College, Cambridge, Gresham Professor of Geo- 
 metry in the City of London, and afterwards 
 Savilian Professor of Geometry at Oxford, to 
 whose work as the successor of Napier in the task 
 of construction of logarithmic tables in an improved 
 form I shall have to refer later. In a letter 
 to Archbishop Ussher, dated Gresham House, 
 March 10, 161 5, Briggs wrote, " Napper, lord of 
 Markinston, hath set my head & hands a work 
 with his new & 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." Briggs visited Napier, and stayed 
 
 H.
 
 1 8 JOHN NAPIER 
 
 with him a month in 1615, again visited him in 
 1 61 6, and intended to visit him again in 16 17, 
 had Napier's life been spared. Another eminent 
 English Mathematician, Edward Wright, a Fellow 
 of Gonville and Caius College, who at once saw 
 the importance of logarithms in connection with 
 navigation, in the history of which he occupies a 
 conspicuous place, translated the " Descriptio," 
 but died in 161 5 before it could be published. 
 The translation was however published in 1618 
 by his son Samuel Wright. 
 
 The contents of the " Descriptio " and of the 
 " Construction 
 
 The " Descriptio " consists of an ornamental 
 title page, fifty-seven pages of explanatory matter, 
 and ninety pages of tables. A specimen page of 
 the tables is here reproduced. The explanatory 
 matter contains an account of Napier's conception 
 of a logarithm, and of the principal properties of 
 logarithms, and also of their application in the
 
 Gr. 
 
 P 
 
 fnin\ 
 
 + 1 — 
 
 Smut } Logarithmi \ Differentia \ U^arithmi f | Sinus f 
 
 1*6434? 
 1*67218 
 1570091 
 
 1572964 
 15787°? 
 
 18**1174 
 18532826 
 
 18514*11' 
 
 18427295 
 18408484 
 
 18389707 
 
 1581*81 
 1*84453 
 15873** 
 
 1590197 
 1593069 
 I59594I 
 
 1849*23* 
 18477984 
 1 8459772 
 
 18441594 
 18423451 
 18405341 
 
 113*81 
 
 124342 
 124804 
 
 18387265 
 18369223 
 18351214 
 
 j 8 370964 
 183*2253 
 i83>35 76 
 
 183H933 
 18296324 
 
 18277747 
 
 125267 
 12*731 
 1 26 196 
 
 18259203 
 18240692 
 18222213 
 
 126662 
 127127 
 
 i"-7*94 
 
 9876882 
 
 5876427 
 9875971 
 
 9875514 
 987*056 
 
 9874597 
 
 0874137 
 
 9873677 
 9873216 
 
 1598812 
 1601684 
 1604555 
 
 1607426 
 1610297 
 1613168 
 
 1616038 
 1618909 
 1621779 
 
 18333237 
 18315294 
 182^7384 
 
 18279*07 
 18261663 
 182438*1 
 
 18226071 
 18208323 
 18190606 
 
 18203765 
 1818*3*1 
 18166969 
 
 18148619 
 18130301 
 18112014 
 
 128062 
 128*31 
 129061 
 
 ? 29472 
 129943 
 130415 
 
 60 
 59 
 _** 
 
 57 
 56 
 
 jj 
 
 *4 
 
 *3 
 
 *2 
 
 130888] 
 1 3 1 3 62 
 131837 
 
 18093758 
 
 1807*533 
 180573 z 8 
 
 1624649 
 1627*19 
 1630389 
 
 76 33~259 
 1636129 
 
 1638999 
 
 18172924 
 1815*273 
 
 18137654 
 18120067 
 18102*11 
 
 18084987 
 
 13*313 
 132790 
 133268 
 
 18039177 
 18021047 
 18002948 
 
 17984880 
 
 17966842 
 
 17948835 
 
 133747 
 
 13-4226 
 134706 
 
 13*187 
 13*669 
 136152 
 
 9871362 
 
 9870897 
 98704^1 
 
 9869964 
 9869496 
 9869027 1 
 
 51 
 
 50 
 
 9872754 
 9872291 
 
 9871827 J 49 
 48 
 
 47 
 46 
 
 4* 
 44 
 
 43 
 
 9X6^5 57.1 
 9868087 
 9867616 I 
 
 4* 
 
 41 
 43 
 
 9867144 
 9866671 
 9866197 
 
 39 
 38 
 37 
 
 9*65722 
 9865246 
 9864770 
 
 36 
 3* 
 34 
 
 33 
 3* 
 
 u 
 
 30 
 
 ttin 
 
 t 
 
 1641868 
 1644738 
 1647607 
 
 16*0476 
 
 1806749* 
 18050034 
 18032604 
 
 1801*207 
 
 I K 
 
 \ 17930859 
 17912913 
 17894997 
 T7I77TT4 
 
 136636 
 137121 
 137607 
 
 138093 
 
 9864293 
 9863815 
 9863336 
 
 98628*6 
 
 80
 
 JOHN NAPIER 19 
 
 solution of plane and spherical triangles. Napier's 
 well-known rules of circular parts containing the 
 complete system of formulae for the solution of 
 right-angled spherical triangles are here given. 
 The logarithms given in the tables are those of 
 the sines of angles from o° to 90 at intervals of 
 one minute, to seven or eight figures. The table 
 is arranged semi-quadrantally, so that the loga- 
 rithms of the sine and the cosine of an angle 
 appear on the same line, their difference being 
 given in the table of differentiae which thus forms 
 a table of logarithmic tangents. It must be 
 remembered that, at that time and long after- 
 wards, the sine of an angle was not regarded, as 
 at present, as a ratio, but as the length of that 
 semi-chord of a circle of given radius which sub- 
 tends the angle at the centre. Napier took the 
 radius to consist of io 7 units, and thus the sine of 
 90°, called the whole sine, is io v ; the sines of 
 smaller angles decreasing from this value to zero. 
 The table is therefore one of the logarithms of 
 
 2 — 2
 
 2o JOHN NAPIER 
 
 numbers between io 7 and o, not for equidistant 
 numbers, but for the numbers corresponding to 
 equidistant angles. It is important to observe 
 that the logarithms in Napier's tables are not 
 what we now know under the name of Napierian 
 or natural logarithms, i.e. logarithms to the 
 base e. His logarithms are more closely related 
 to those to the base i/e ; the exact relation 
 
 Nap 
 
 is that, if x is a number, and Log x its 
 logarithm in accordance with Napier's tables, 
 
 Nap 
 T qqt % . X 
 
 — ^— is the logarithm of — - t to the base 
 io 7 b io 7 
 
 lie ; thus 
 
 Nap 
 , > Log.* 
 X /l V 
 
 io' 
 
 io' \ej 
 
 Nap 
 
 or Log x = i o 7 log e i o 7 — i o 7 log e x. 
 
 Napier had no explicit knowledge of the exist- 
 ence of the number £, nor of the notion of the 
 base of a system of logarithms, although as we 
 shall see he was fully cognizant of the arbitrary
 
 JOHN NAPIER 21 
 
 element in the possible systems of logarithms. 
 His choice was made with a view to making the 
 logarithms of the sines of angles between o° and 
 90 , i.e. of numbers between o and io 7 , positive 
 and so as to contain a considerable integral part 
 
 The " Constructio " consists of a preface of 
 two pages, and fifty-seven pages of text. The 
 conception of a logarithm is here clearly explained, 
 and a full account is given of the successive steps 
 by which the Canon was actually constructed. In 
 this work one of the four formulae for the solution 
 of spherical triangles, known as Napier's analogies, 
 is given, expressed in words ; the other three 
 formulae were afterwards added by Briggs, being 
 easily deducible from Napier's results. 
 
 The decimal point. 
 
 Our present notation for numbers with the 
 
 decimal point appears to have been independently 
 
 invented by Napier, although a point or a half 
 
 bracket is said to have been employed somewhat
 
 22 JOHN NAPIER 
 
 earlier by Jobst Biirgi for the purpose of separat- 
 ing the decimal places from the integral part of a 
 number*. The invention of decimal fractions was 
 due to Simon Stevin (i 548-1 620), who published 
 a tract in Dutch, "De Thiende," in 1585, and in 
 the same year one in French under the title " La 
 Disme," in which the system of decimal fractions 
 was introduced, and in which a decimal system of 
 weights, measures and coinage was recommended. 
 In the "Rabdologia" Napier refers to Stevin in an 
 '* Admonitio pro Decimali Arithmetica " in which 
 he emphasizes the simplification arising from the 
 use of decimals, and introduces the notation with 
 the decimal point. By Stevin and others the 
 notation 94© 1 ©3 ®o® 5 or 94 i'fo'"f", 
 for example, was used instead of Napier's notation 
 94*1305. Later on Briggs sometimes used the 
 notation 941305. It is clear that the notation 
 introduced by Napier, which was however not 
 
 * The decimal point was also employed by Pitiscus in the 
 tables appended to the later edition, published in 1612, of his 
 " Trigonometria."
 
 JOHN NAPIER 23 
 
 universally adopted until the eighteenth century, 
 was far better adapted than the more complicated 
 notation used by Stevin and later writers to bring 
 out the complete parity of the integral and decimal 
 parts of a number in relation to the operations of 
 arithmetic, and to emphasize the fact that the 
 system of decimal fractions involves only an 
 extension of the fundamental conception of our 
 system of notation for integral numbers, that the 
 value of a digit, in relation to the decimal scale, is 
 completely indicated by its position. 
 
 Napier s definition of a logarithm. 
 
 1 1 
 A T 
 
 1 111 
 
 a AAA 
 
 1 
 s 
 
 ft T x 
 
 & Q2 G & 
 
 tO CO 
 
 Napier supposes that on a straight line TS 
 which he takes to consist of io 7 units, the radius 
 of the circle for which the sines are measured, a 
 point P moves from left to right so that its velocity 
 is at every point proportional to the distance from
 
 24 JOHN NAPIER 
 
 S. He supposes that on another straight line a 
 point Q moves with uniform velocity equal to that 
 which P has when at T, and that Q is at T lt when 
 P is at T. When P has any particular position 
 Pj in the course of its motion, the logarithm of the 
 sine or length SP 1 is defined to be the number 
 representing the length T X Q X , from T x to the 
 position <2i of Q at the time when P is at P x , 
 Thus the logarithm of the whole sine TS(= io 7 ) 
 is o, and the logarithm of any sine less than io 7 
 is positive and increases indefinitely as the sine 
 diminishes to zero. Napier recognized that in 
 accordance with this definition, the logarithm of 
 a number greater than io 7 , corresponding to Sp lf 
 where p 1 is the position of P when it has not yet 
 reached T, will be negative, the corresponding 
 position of Q being on the left of T x . 
 
 Let Q lt Q 2 , Q 3 , <2 4 , ... be a number of positions 
 of Q such that Q 1 Q i = Q 2 Q 3 =Q 3 Q i = ... and let 
 P lt P 2> P 3 , ... be the corresponding positions of 
 P; so that P X P^ P 2 P 3 , P%P» ••• are spaces
 
 JOHN NAPIER 25 
 
 described by P in equal times. Napier then 
 shews by means of special illustrations that 
 
 and thus that, corresponding to a series of values, 
 of T^Q that are in arithmetic progression, there 
 are a series of values of SP that are in geometric 
 progression. 
 
 The matter may be put in a concise form 
 which represents the gist of Napier's reasoning, 
 and of the essential point of which he had a clear 
 intuition. 
 
 P P' 
 
 1 > r-J 1 . 
 
 PS PS 
 Let -p-* = -^-p. ; and let p be any point 
 
 in P x P it and /' the corresponding point in P t P, ; 
 so that P,p : pP^P^p' : p'P 3 . The velocity of 
 the moving point when at/ bears a constant ratio 
 
 ^=^=-~-p) to its velocity when at /. 
 
 As this holds for every corresponding pair of
 
 26 JOHN NAPIER 
 
 points p,p' in the two intervals P X P» P 2 P 3 , it is 
 
 dear that the motion in P^P % takes place in the 
 
 same time as that in PJ\ ; the velocities at all 
 
 corresponding points being changed in the same 
 
 ratio, that of P x P t to P 2 P a . Hence the result 
 
 follows that 
 
 P 1 S_P 2 S_P 9 S_ 
 P t S m P t S m P t S m " u 
 
 if the points Q lf <2 2 , Q 3 , Q 4 , ... are such that 
 QiQ*=Q*Q*=Q%Qi=->\ i.e. if the spaces P x P %t 
 P 2 P 3 , P 3 P 4 , ... are described in equal times. Thus 
 the logarithms of a set of numbers in geometric 
 progression are themselves in arithmetic pro- 
 gression. 
 
 In our modern notation, if x = SP, we have 
 
 — = r, where V denotes the velocity of P 
 
 dt io 7 
 
 at T; and if V—'T.Q, -Z=V\ thus -y- = =; 
 
 * dt dy io 7 
 
 and accordingly Napier's method amounts to an 
 
 intuitional representation of the integration of this 
 
 differential equation.
 
 JOHN NAPIER 27 
 
 The limits of a logarithm. 
 
 As no method was available by which a 
 logarithm could be calculated to an arbitrarily 
 great degree of approximation, Napier obtained 
 two limits between which a logarithm must lie, 
 and his whole method of construction depends 
 upon the use of these limits, together with corre- 
 sponding limits for the value of the difference of 
 the logarithms of two numbers. 
 
 — 1 1 1 
 
 T P x S 
 
 — 1 1 1 
 
 ft T, G 
 
 Since the velocities of P and Q at T, T x are 
 the same, and the velocity of P decreases after- 
 wards whereas that of Q remains constant, it is 
 clear that TP l < T^Q^. Again let p t T on the left 
 of T be described in the same time as TP Xi so 
 that ^7", = 7\ Q v It is then clear that p l T>q l T l .
 
 28 JOHN NAPIER 
 
 Nap 
 
 If x = P 1 S, hogx= T X Q„ we thus have 
 
 Nap 
 
 hogx> TP lt or jo 7 —x; 
 
 Nap IQ 7 
 
 and Log x=y 1 T 1 <f> 1 T, or TP 1 — -, which is 
 
 (io 7 -*) — . Thus 
 
 ' x 
 
 j Q 7 Nap 
 
 (io 7 — x) — >Logx> io 7 —x ...(i); 
 
 these are Napier's limits for a logarithm. 
 In a similar manner it is shewn that 
 
 y—x Nap Nap y — x 
 \o- > hogx— Logy> io 7 - ...(2), 
 
 where x<y; these are the limits which Napier 
 
 Nap Nap 
 
 employs for Log x — Log jj/. 
 
 The results (1) and (2) were given by Napier 
 in words, no short designation being employed 
 even for a logarithm. 
 
 Napier s construction of the canon. 
 
 The first step taken by Napier in the process 
 of constructing the canon was to form three tables
 
 JOHN NAPIER 29 
 
 of numbers in geometric progression. The first 
 table consists of 101 numbers, of which io 7 is the 
 
 first, and of which 1 = is the common ratio ; 
 
 io 7 
 
 thus (in modern notation) the table consists of the 
 
 numbers io 7 ( 1 ; ) , where r has the values o to 
 
 \ io 7 / 
 
 100. Each number was formed by subtracting 
 
 from the preceding one the number obtained by 
 
 moving the digits seven places to the right. 
 
 First Table 
 X°V~itf)> '-oto iooj 
 
 I OOOOOOO'OOOOOOO 
 I'OOOOOOO 
 
 99999990000000 
 
 -9999999 
 
 9999998 'OOOOOO I 
 
 '9999998 
 
 999999 7 '0000003 
 
 '9999997 
 
 9999996*0000006 
 
 to be continued up to 
 
 9999900*0004950
 
 30 JOHN NAPIER 
 
 The second table consists of the 51 numbers 
 io 7 (i 1) , where r = o, 1, ... 50. The com- 
 mon ratio 1 1 is nearly equal to ( 1 7 J , 
 
 that of the last number in the first table, to the 
 first in that table. 
 
 Second Table 
 
 { Io7 ( x -^) r ' "'—»*} 
 
 I OOOOOOO'OOOOOO 
 IOO'OOOOOO 
 
 9999900*000000 
 99*999000 
 
 9999800*001000 
 
 to be continued up to 
 
 9995001*222927 
 
 In this table there is an arithmetical error, as 
 the last number should be 9995001*224804; the 
 effect of this error on the canon will be referred 
 to later. 
 
 The ratio of the last number to the first is 
 
 1 N 50 
 1 " loV ' Whkh is nearly I ~ 2oW'
 
 JOHN NAPIER 31 
 
 The third table consists of 69 columns, and 
 each column contains 21 numbers. The first 
 number in any column is obtained by taking 
 1 — y^ of the first number in the preceding 
 column. The numbers in any one column are 
 obtained by successive multiplication by 1 — 275V0 ; 
 thus the/^ number in the q th column is 
 
 First column 
 
 I OOOOOOO "oooo 
 
 9995000*0000 
 
 9990002*5000 
 
 9985007*4987 
 
 continued to 
 9900473-5780 
 
 Third Table 
 
 Second column 
 
 9900000-0000 . 
 
 9895050*0000 . 
 
 9890102-4750 . 
 
 9885157-4237 . 
 
 continued to 
 
 69th column 
 5048858*8900 
 5046334-4605 
 5043811-2932 
 5041289-3879 
 
 continued to 
 4998609-4034 
 
 9801468-8423 
 
 The ratio of the last to the first number in 
 any one column is ( 1 — ^uVo") 20 , which is nearly 
 
 i-t4tt or i^fr- 
 
 It will be observed that the last number of the 
 
 last column is less than half the radius, and thus 
 
 corresponds to the sine of an angle somewhat less 
 
 than 30°.
 
 32 JOHN NAPIER 
 
 In this table there are, speaking roughly, 
 68 numbers in the ratio ioo : 99 interpolated 
 between io 7 and \\d ; and between each of these 
 are interpolated twenty numbers in the ratio 
 10000 : 9995. 
 
 Having formed these tables, Napier proceeds 
 to obtain with sufficient approximation the loga- 
 rithms of the numbers in the tables. For this 
 purpose his theorems (1) and (2) as to the limits 
 of logarithms are sufficient. In the first table, the 
 logarithm of 9999999 is, in accordance with (1), 
 between rooooooi and I'ooooooo, and Napier 
 takes the arithmetic mean 1*00000005 f° r tne 
 required logarithm. The logarithm of the next 
 sine in the table is between 2 "0000002 and 
 2 •0000000; for this he takes 2-00000010, for 
 the next sine 3*00000015, and so on. 
 
 The theorem (2) is used to obtain limits for 
 the logarithms of numbers nearly equal to a 
 number in the first table. In this way the loga- 
 rithm of 9999900 the second number in the second
 
 JOHN NAPIER 33 
 
 table is found to lie between 100*0005050 and 
 100*0004950; the next logarithm has limits double 
 of these, and so on. The logarithm of the last 
 sine in the second table is thus found to lie 
 between 5000*0252500 and 5000*0247500. The 
 logarithms of the numbers in the second table 
 having thus been found to a sufficient degree of 
 approximation, the logarithm of a number near 
 one in the second table is found thus : — Let y 
 be the given sine, x the nearest sine in the table ; 
 
 say y < x. Determine z so that — ; = - , then 
 
 J y 10 7 x 
 
 Nap Nap Nap Nap Nap 
 
 Log g = Log z — Log io 7 = Logjj/ — Log .a: ; 
 
 Nap 
 
 find the limits of Logz by means of the first table, 
 and when these are found add them to those of 
 
 Nap Nap 
 
 Log:r, and we thus get the limits of Logy. In 
 this manner limits are found for the logarithms 
 of all the numbers in the first column of the 
 third table; thus those of 9900473*57808 are 
 100024*9657720 and 100024*9757760, and the 
 h. 3
 
 34 JOHN NAPIER 
 
 logarithm is taken to be 100024*9707740, the 
 mean of the two limits. The first number in 
 the second column differs only in the fifth cypher 
 from the last number in the first column, and thus 
 its logarithm can be calculated approximately. 
 The logarithms of all the other numbers in the 
 table can then be found, since the logarithms of 
 all the numbers in any one column, or in any one 
 row, are in arithmetic progression. 
 
 When the logarithms of all the numbers in 
 the third table have thus been calculated, the 
 table formed by filling them in is called by 
 Napier his radical table, and is of the form 
 given on the opposite page. 
 
 The radical table being completed, the loga- 
 rithms in it are employed for the calculation of 
 the principal table or canon. For this purpose 
 the logarithms of sines very nearly equal to the 
 whole sine io 7 are obtained simply by subtracting 
 the given sine from io ? . The logarithm of a sine 
 embraced within the limits of the radical table is
 
 
 <3 
 
 
 t/i 
 
 00 m CONO 
 
 00 
 
 
 S 
 
 .3 
 
 lO t"»00 ON 
 
 b 
 
 
 CM N CM N 
 
 to 
 
 
 
 CM N CM CM 
 
 CM 
 
 
 A 
 
 rj- On tJ- On 
 
 * 
 
 
 ro co •* tj- 
 
 CO 
 
 
 O 
 
 00 00 00 00 
 
 ON 
 
 c 
 
 O O VO VO 
 
 NO 
 
 § 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 ■ 
 
 O to CM On 
 
 * 
 
 ■a 
 
 V 
 
 O O co t^- 
 
 CO 
 
 On 
 
 ifl 
 
 OnnO OnOO 
 
 O 
 
 NO 
 
 s 
 
 00 "t CM CO 
 
 *t 
 
 
 c 
 
 oo tj- h b\ 
 
 On 
 
 
 
 u-j co <■* O0 
 
 O 
 
 
 "rt 
 
 00 COOO N 
 
 NO 
 
 
 E 
 
 3 
 
 00O «H 
 
 00 
 
 
 * "t •* ^ 
 
 ON 
 
 
 Id 
 
 o o o o 
 
 On 
 
 
 £ 
 
 IfllOmifl 
 
 "* 
 
 
 i 
 
 piO CO H 
 
 CM 
 
 
 fO 't lO t>- 
 
 00 
 
 
 
 o o o o 
 
 M 
 
 
 
 iO u-> U-) IT) 
 
 to 
 
 
 O in O to 
 
 o 
 
 C 
 
 o 
 
 O O M M 
 
 o 
 
 J 
 
 M tH M M 
 
 CM 
 
 
 
 
 <J 
 
 
 
 
 -o 
 
 E 
 
 O O O t- 
 
 CO 
 
 c 
 
 J 
 
 O O to CO 
 
 CM 
 
 o 
 
 O O •* cm 
 
 •f 
 
 1 
 
 e 
 
 O O J» Tf 
 
 00 
 
 w 
 
 3 
 
 c 
 
 Q O H t« 
 
 00 
 
 
 
 O to O to 
 
 NO 
 
 
 "2 
 
 2 
 
 o O •■* •■* 
 
 ■* 
 
 
 O to O to 
 
 
 
 O ON OnOO 
 
 o 
 
 
 a 
 
 OnOO 00 00 
 
 00 
 
 
 £ 
 
 ON On On On 
 
 On 
 
 
 !/) 
 
 
 
 
 E 
 
 O cm yo r-» 
 
 O 
 
 
 •C 
 
 m cm co 
 
 io 
 
 
 °C 
 
 O O O 
 
 CM 
 
 
 I 
 
 O O O 
 
 O 
 
 
 to O to 
 
 o 
 
 a 
 
 M M 
 
 o 
 
 t-l 
 
 g 
 
 J3 
 
 
 
 
 
 
 
 D 
 
 
 
 
 i2 
 
 o o o ^ 
 
 o 
 
 ^j 
 
 £ 
 
 O O O 00 
 
 00 
 
 2 
 
 £ 
 
 O O O On 
 
 r- 
 
 O O to tJ- 
 
 JO 
 
 3 
 
 s 
 
 o o cm r* 
 
 CO 
 
 
 
 o o o o 
 
 t^» 
 
 
 1 
 
 o o o o 
 
 ■^- 
 
 
 O to O to 
 
 o 
 
 
 O On OnOO 
 
 o 
 
 
 ■ 
 
 55 
 
 O On ON On 
 
 ON 
 
 
 O On On On 
 
 ON 
 
 
 
 M 
 
 
 3—2
 
 36 
 
 JOHN NAPIER 
 
 found thus : — Let x be the given sine, y the 
 nearest sine in the table ; say x >y ; calculate 
 (x— y) io ? and divide it either by x or by y, or 
 by some number between the two ; then add the 
 result to the logarithm of the table sine. 
 
 For the purpose of finding the logarithm of a 
 sine which is not embraced within the limits of 
 the radical table, Napier gives a short table 
 containing the difference of the logarithms of two 
 sines of which the ratio is compounded of the 
 ratios 2 : 1, and 10 : 1. 
 
 Short Table 
 
 Given 
 
 Corresponding 
 
 Given 
 
 Corresponding 
 
 proportion 
 
 difference of 
 
 proportion 
 
 difference of 
 
 of sines 
 
 logarithms 
 
 of sines 
 
 logarithms 
 
 2 
 
 to I 
 
 6931469-22 
 
 8000 to 
 
 I 89871934*68 
 
 4 
 
 >J 
 
 13862938-44 
 
 I OOOO , 
 
 , 92103369-36 
 
 8 
 
 51 
 
 20794407*66 
 
 20000 , 
 
 , 99034838-58 
 
 10 
 
 J> 
 
 23025842-34 
 
 40000 , 
 
 , 105966307*80 
 
 20 
 
 » 
 
 2995731156 
 
 80000 , 
 
 , 112897777*02 
 
 40 
 
 » 
 
 36888780-78 
 
 I OOOOO , 
 
 , 11512921170 
 
 80 
 
 >) 
 
 43820250-00 
 
 200000 , 
 
 , 122060680-92 
 
 100 
 
 » 
 
 46051684-68 
 
 400000 , 
 
 , 128992150*14 
 
 200 
 
 »» 
 
 52983153-90 
 
 800000 , 
 
 1 i359 2 3 6l 9"36 
 
 400 
 
 » 
 
 59914623-12 
 
 I OOOOOO , 
 
 • i38i55°54 , o4 
 
 800 
 
 >> 
 
 66846092-34 
 
 2000000 , 
 
 , 145086523-26 
 
 1 000 
 
 H 
 
 69077527*02 
 
 4000000 , 
 
 , 152017992-48 
 
 2000 
 
 » 
 
 76008996*24 
 
 8000000 , 
 
 , 158949461-70 
 
 4000 
 
 M 
 
 82940465-46 
 
 lOOOOOOO , 
 
 , 161180896-38
 
 JOHN NAPIER 37 
 
 Nap 
 
 To calculate this table, Napier found Log io 7 
 
 Nap 
 
 and Log 500000 by using the radical table ; and 
 thus 6931469*22 was found as the difference of 
 the logarithms of numbers in the ratio 2:1. The 
 difference of logarithms of sines in the ratio 8 : 1 
 is three times 6931469*22, i.e. 20794407*66. The 
 sine 8000000 is found by using the radical table 
 to have for its logarithm 2231434*68, whence by 
 addition the logarithm of the sine 1000000 is 
 found to be 23025842*34. Since the radius is ten 
 times this sine, all sines in the ratio 10 : 1 will 
 have this number for the difference of their 
 logarithms. The rest of the table was then 
 calculated from these determinations. 
 
 The logarithms of all sines that are outside 
 the limits of the radical table could now be deter- 
 mined. Multiply the given sine by 2, 4, 8, ... 
 200, ... or by any proportional number in the 
 short table, until a number within the limits of 
 the radical table is found. Find the logarithm
 
 38 JOHN NAPIER 
 
 of the sine given by the radical table, and add to 
 it the difference which the short table indicates. 
 
 In the manner described, the logarithms of 
 all sines of angles between o° and 45 s could now 
 be determined, and the principal table or canon 
 completed. Napier gave, however, a rule by 
 which when the logarithms for all the angles not 
 less than 45 ° are known, the logarithms for all 
 the angles less than 45 ° can be determined. This 
 rule we may write in the form 
 
 Nap Nap 
 
 Log \ io 7 + Log sin x 
 
 Nap Nap 
 
 = Log sin \x+ Log sin (90 — \x), 
 
 cos ^-x & 
 
 which follows from the fact that — — — = . 2 , 
 
 sin x sin \x 
 
 when we take into account the change in the 
 
 definition of a sine. 
 
 The accuracy of Napier s Canon. 
 It has been observed above that a numerical 
 error occurs in the value of the last number in 
 the second table. As Napier employed this
 
 JOHN NAPIER 39 
 
 inaccurate value in his further calculations, it 
 produced an error in the greater part of his 
 logarithmic tables. The effect of this error is 
 that most of the logarithms are diminished by 
 about J io~ 6 of their correct values. Napier 
 himself observes in the " Constructio" that some 
 of the logarithms he obtained by means of his rule 
 for finding the logarithms of numbers outside the 
 limits of the radical table differ in value from 
 the logarithms of the same numbers found by the 
 rule for determining the logarithms of sines of 
 angles less than 45 °. He attributes this dis- 
 crepancy to defects in the values of the natural 
 sines he employed, and suggested a recalculation 
 of natural sines in which io 8 should be the radius. 
 Owing to these two causes, the last figure in the 
 logarithms of the canon is not always correct. 
 
 The improved system of logarithms. 
 
 The special purpose of application to trigono- 
 metrical calculations accounts for Napier's choice
 
 40 JOHN NAPIER 
 
 of the system in which the logarithm of io 7 is 
 zero, and the logarithms of sines of angles between 
 o° and 90° are positive. It is, however, clear 
 that the rule of the equality of the sum of the 
 logarithms of two numbers and that of their pro- 
 duct would hold for numbers in general, only 
 if the logarithm of unity were taken to be zero, 
 as a number is unaltered by multiplication by 
 unity. On this account, Napier, in an appendix 
 to the " Constructio," proposed the calculation of 
 a system of logarithms in which Log 1 = o, and 
 Logio=io 10 . This is practically equivalent to 
 the assumption Log 10= 1, as the former assump- 
 tion merely indicates that the logarithms are to 
 be calculated to 10 places of decimals. Briggs 
 pointed out, in his lectures at Gresham College, 
 that a system would be convenient, on .which 
 o should be the logarithm of 1, and io 10 that of the 
 tenth part of the whole sine (viz. sin 5 44/ 21 "), 
 which would be equivalent to Log^= io 10 . This 
 system he suggested to Napier during his visit
 
 JOHN NAPIER 41 
 
 to Merchiston in 161 5, when Napier pointed out 
 that the same idea had occurred to himself, but 
 that the assumption Log 10= io 10 would lead to 
 the most convenient system of all, and this was 
 at once admitted by Briggs. 
 
 In the appendix above referred to, Napier 
 gives some indications of methods by which the 
 improved logarithms might be calculated. These 
 depend upon exceeding laborious successive ex- 
 tractions of fifth and of square roots, which work 
 he proposed should be carried out by others, 
 and especially by Briggs. In an "Admonitio" 
 printed in the " Constructio," Napier remarked 
 that it is a matter of free choice to what sine or 
 number the logarithm o is assigned, that it is 
 necessary frequently to multiply or divide by 
 the complete sine (sin 90°), and thus that a 
 saving of trouble arises if the logarithm of this 
 sine be taken to be zero. 
 
 Briggs immediately set about the calculation 
 of these improved logarithms, and in the following
 
 42 JOHN NAPIER 
 
 year, when he again visited Napier, shewed him 
 a large part of the table which was afterwards 
 published in 1624. On the death of Napier in 
 161 7 the whole work of developing the new 
 invention passed into the skilful hands of Briggs, 
 who, in the same year, published his " Logarith- 
 morum Chilias Prima," containing the common or 
 Briggian logarithms of the first thousand numbers 
 to 14 places of decimals. In 1624 he pub- 
 lished the "Arithmetica Logarithmica," a table of 
 logarithms of the first 20000 numbers and of the 
 numbers from 90000 to 100000, to 14 places of 
 decimals. The gap between 20000 and 90000 
 was fitted up by Adrian Vlacq, who published 
 in 1628 at Gouda a table of common logarithms 
 of numbers from 1 to 1 00000, to 10 places of 
 decimals. Vlacq's tables, although not free from 
 error, have formed the basis of all the numerous 
 tables of logarithms of natural numbers that have 
 been since published.
 
 JOHN NAPIER 43 
 
 Other Tables. 
 
 A table of logarithms exactly similar to those 
 of Napier in|the " Constructio " was published in 
 1624 by Benjamin Ursinus at Cologne. The 
 intervals of the angles are 10", and the logarithms 
 are given to 8 places. The first logarithms to 
 the base e were published by John Speidell in his 
 "New Logarithmes," in London in 1619; this table 
 contains logarithmic sines, tangents and secants 
 for every minute of the quadrant to 5 decimal 
 places. 
 
 Predecessors of Napier. 
 
 It is usually the case that the fundamental 
 conceptions involved in a great new invention 
 have a history, which reaches back to a time, 
 often a long time, before that of the inventor. 
 Although Napier's introduction of logarithms is 
 justly entitled to be regarded as a really new
 
 44 JOHN NAPIER 
 
 invention, it is not an exception to the usual rule. 
 The notion of an integral power of a ratio was 
 employed by the Greek Mathematicians. The 
 nature of the correspondence between a geometric 
 progression and an arithmetic progression had 
 been observed by various Mathematicians. In 
 particular Michael Stifel (i 486-1 567), in his 
 celebrated " Arithmetica Integra," published in 
 1544, expressly indicated the relations between 
 operations with the terms of a geometric and an 
 arithmetic series, in which the terms are made to 
 correspond, viz. the relations between multiplica- 
 tion, division and exponentiation on the one 
 hand, and addition, subtraction and multiplication 
 or division by an integer on the other hand. 
 But no indication was given by Stifel or others 
 how this correspondence could be utilized for 
 the purpose of carrying out difficult arithmetical 
 calculations. There were even given by the 
 Belgian Mathematician Simon Stevin (1548- 
 1620) certain special tables for the calculation
 
 JOHN NAPIER 45 
 
 of interest, consisting of tables of the values of 
 
 ff , and of — V -, — - — ej + ... + 
 
 (i+r) n ' i+r (i+rf (i+r) n * 
 
 The first of these tables are really tables of 
 antilogarithms, but there were given no theoretical 
 explanations which would extend the use of the 
 tables beyond their special purpose. Napier, 
 whether he was acquainted with Stifel's work 
 or not, was the first whose insight enabled him 
 to develop the theoretical relations between 
 geometric and arithmetic series into a method 
 of the most far-reaching importance in regard 
 to arithmetic calculations in general. On the 
 theoretical side, Napier's representation by con- 
 tinuously moving points involved the conception 
 of a functional relationship between two con- 
 tinuous variables, whereas Stifel and others 
 had merely considered the relationship between 
 two discrete sets of numbers. This was in itself 
 a step of the greatest importance in the develop- 
 ment of Mathematical Analysis.
 
 46 JOHN NAPIER 
 
 A rival inventor. 
 
 No account of the invention of logarithms 
 would be complete without some reference to 
 the work of Jobst Biirgi (i 552-1632), a Swiss 
 watch-maker and instrument-maker, who in- 
 dependently invented a system of logarithms. 
 His system was published in 1620, after Napier's 
 Canon had become known and fully recognized, 
 in a work entitled "Arithmetische und Geo- 
 metrische Progress-Tabulen." The table is really 
 an antilogarithmic table, and consists of a set of 
 numbers printed red placed in correspondence 
 with a set of numbers printed black. The red 
 numbers are o, 10, 20, ... , those of an arithmetic 
 series, and the corresponding black numbers are 
 1 00000000, 1 000 1 0000, 1 0002000 1, of a geometric 
 series ; thus the red numbers are the logarithms of 
 the black ones divided by io 8 with the base 
 
 y i*O0Ol. Biirgi appears to have devised his
 
 JOHN NAPIER 47 
 
 system a good many years before he published it, 
 but kept it secret until he published his tables 
 six years after the appearance of those of 
 Napier. 
 
 Conclusion. 
 
 The system of Biirgi is decidedly inferior to 
 that of Napier, and the knowledge of the use of 
 logarithms which was spread in the scientific 
 world was entirely due to the work of Napier. 
 
 The concensus of opinion among men of 
 Science of all nations has ascribed to Napier 
 the full honour due to the inventor of the method 
 which has provided the modern world with a tool 
 that is indispensable for all elaborate arithmetical 
 calculations. In the great advance which had 
 taken place in Mathematical Science during the 
 half century preceding the publication of the 
 " Constructio," British Mathematicians had taken 
 no part. It is very remarkable that, in a country
 
 48 JOHN NAPIER 
 
 distracted by political, social, and religious feuds 
 of the most serious kind, such as Scotland then 
 was, there should have arisen the first of those 
 great thinkers who in the course of the seven- 
 teenth century brought Great Britain to the 
 highest point of achievement in the domain of 
 Mathematical Science. 
 
 CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS
 
 UC SOUTHERN REGIONAL LIBRARY FACILITY 
 
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