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life tt<N John Napier and the Invention of Logarithms, 1614 CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, Manager ILotrtJOtt: FETTER LANE, E.C. ii&mburgf): ioo PRINCES STREET Berlin: A. ASHER AND CO. leip>ifl: F. A. BROCKHAUS $tio Sot*: G. P. PUTNAM'S SONS Bombag ano Calcutta: MACMILLAN AND CO., Ltd. Toronto: J. M. DENT AND SONS, Ltd. STofcso : THE MARUZEN-KABUSHIKI-KAISHA All rights reserved ~~h2j°^- m-OWw- <>£^- y *&'<rfy\/bw 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 AA 000139 312 3