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THE

CONSTRUCTION OF LOGARITHMS

WITH A

CATALOGUE OF NAPIER'S WORKS

a 2

THE CONSTRUCTION OF THE WONDERFUL CANON OF

LOGARITHMS

BY

JOHN NAPIER

BARON OF MERCHISTON

TRANSLATED FROM LATIN INTO ENGLISH WITH NOTES

AND

A CATALOGUE

OF THE VARIOUS EDITIONS OF NAPIER'S WORKS, BY WILLIAM RAE MACDONALD, F.F.A.

WILLIAM BLACKWOOD AND SONS

EDINBURGH AND LONDON

MDCCCLXXXIX

•2)3 A/ 2)52.

1

I

/I

'> \

To

The Right Honourable

FRANCIS BARON NAPIER AND ETTRICK, K,T,

descendant of

John Napier of Merchiston

this Translation of the

MirificI Logarithmorum Canonis Constructio

is dedicated with much respect.

b 2

CONTENTS.

INTRODUCTION, .......

THE CONSTRUCTION OF LOGARITHMS, BY JOHN NAPIER, PREFACE BY ROBERT NAPIER, THE CONSTRUCTION,

APPENDIX, ..... REMARKS ON APPENDIX BY HENRY BRIGGS, TRIGONOMETRICAL PROPOSITIONS, . NOTES ON TRIGONOMETRICAL PROPOSITIONS BY HENRY BRIGGS,

NOTES BY THE TRANSLATOR, . . .

A CATALOGUE OF THE WORKS OF JOHN NAPIER,

PRELIMINARY, ..... THE CATALOGUE, ..... APPENDIX TO CATALOGUE, .... SUMMARY OF CATALOGUE AND APPENDIX, .

PAGE

xi

3 7

48 55 64 76

83

lOI

103 109 148 166

b 3

npKMMMCVH

I I ^^PW^

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INTRODUCTION

John Napier^'' was the eldest son of Archibald Napier and Janet Bothwell. Hejwas born at Merchiston, near Edinburgh, in 1550, when his father could have been little more than sixteen.

Two months previous to the death of his mother, which occurred on 20th December 1 563, he matriculated as a student of St Salvator's College, St Andrews.

JWhile there, his mind was specially directed to the study and searching out of the mysteries of the Apocalypse,

_the xesjiilt of which appeared thirty years later in his first

^published work, ' A plaine discovery of the whole Reve-

-Jation of St John.'

Had he continued at St Andrews, his name would naturally have appeared in the list of determinants for 1566 and of masters of arts for 1568. It is not, how- ever, found with the names of the students who entered college along with him, so that he is believed to havejeft^

* See note, p. 84, as to spelling of name.

b 4 the

xii Introduction.

-the-XLtiiversity .previous to 1566 in order to complete his jstudies on the Continent.

He was at home in 1571 when the preliminaries were arranged for his inarriage with Elizabeth^ daughter of Sir James Stirling of Keir. The marriage took place towards the close of 1572. In 1579 his wife died, leaving him one son, Archibald, who, in 1627, was raised to the peerage by the title of Lord Napier, and also one daughter, Jane.

_A few years afterjthe_death of hi_s first wife he married Agnes, daughter of Sir James Chisholm of Cromlix, who survived him. The offspring of this marriage were five sons and five daughters, the best known of whom is the second son, Robert, his father's literary executor.

Leaving for a moment the purely personal incidents of Napier's life, we may here note the dates of a few of the many exciting public events which occurred during the course of it. In 1560 a Presbyterian form of Church government was established by the Scottish Parliament. On 14th August 1 56 1, Queen Mary, the young widow of Francis II., sailed from Calais, receiving an enthusiastic welcome on her arrival in Edinburgh. Within six years, on 24th July 1567, she was compelled to sign her abdica- tion. The year 1572 was signalised by the Massacre of St Bartholomew, which began on 24th August; exactly three months later, John Knox died. On 8th February 1587 Mary was beheaded at Fotheringay, and in May of the year following the Spanish Armada set sail. The last event we need mention was the death of Queen

Elizabeth

Introduction. xiii

Elizabeth on 24th March 1603, and the accession of King James to the throne of England.

The threatened invasion of the Spanish Armada led Napier to take an active part in Church politics. In 1588 he was chosen by the Presbytery of Edinburgh one of its commissioners to the General Assembly. In October 1593 he was appointed one of a deputation of six to interview the king regarding the punishment of the ** Popish rebels," prominent among whom was his own father-in-law. On the 29th January following, 1594,"^ the letter which forms the dedication to his first publi- cation, * A plaine discovery,' was written to the king.

Not long after this, in July 1594, we find Napier enter- ing into that mysterious contract with Logan of Restalrig for the discovery of hidden treasure at Fast Castle.

Another interesting document written by Napier bears date 7th June 1596, with the title, 'Secrett inuentionis, proffitable & necessary in theis dayes for defence of this Hand & withstanding of strangers enemies of Gods truth and relegion.'

The versatility and practical bent of Napier's mind are further evidenced by his attention to agriculture, which was in a very depressed state, owing to the unsettled con- dition of the country. The Merchiston system of tillage by manuring the land with salt is described in a very rare tract by his eldest son, Archibald, to whom a mono-

* 1593 old style, 1594 new style. Under the old style the year commenced on 25th March.

c poly

xlv Introduction.

poly of the system was granted under the privy seal on 22d June 1598. As Archibald Napier was quite a young man at the time, it is most probable the system was the result of experiments made by his father and grandfather.

About 1603, the Lennox, where Napier held large possessions, was devastated in the conflict between the chief of Macgregor and Colquhoun of Luss, known as the raid of Glenfruin. The chief was entrapped by Argyll, tried, and condemned to death. On the jury which condemned him sat John Napier. The Mac- gregors, driven to desperation, became broken men, and Napier's lands no doubt suffered from their inroads, as we find him on 24th December 161 1 entering into a contract for mutual protection with James Campbell of Lawers, Colin Campbell of Aberuchill, and John Campbell, their brother-german.

To the critical events of 1588 which, as we have already seen, drew Napier into public life, is due the appearance Jn English of [A plaine discovery,' already mentioned. The treatise was intended to have been written in Latin, but, owing to the events above referred to, he was, as he says, ' constrained of compassion, leaving the Latin to haste out in English the present work almost unripe.' It was published in 1594. A revised edition appeared in 1 6 1 1 , wherein he still expressed his intention of rewriting it in Latin, but this was never accomplished.

Mathematics, as well as theology, must have occupied Napier's attention from an early age. What he had done

in

Introduction. xv

in the way of systematising and developing the sciences of arithmetic and algebra, probably some years before the publication of *A plaine discovery/ appears in the manuscript published in 1839 under the title * De Arte Logistica.' From this work it appears that his investi- gations in equations had led him to a consideration of imaginary roots, a subject he refers to as a great algebraic secret. He had also discovered a general method for the extraction of roots of all degrees.

The decimal system of numeration and notation had been introduced into Europe in the tenth century. To complete the system, it still remained to extend the notation to fractions. This was proposed, though in a cumbrous form, by Simon Stevin in 1585, but'i^Napier was the first to use the present notation. "^^'^

Towards the end of the sixteenth century, however, the further progress of science was greatly impeded by the continually increasing complexity and labour of numerical calculation. In consequence of this, Napier seems to have laid aside his work on Arithmetic and Algebra before its completion, and deliberately set himself to devise some means of lessening this labour. By 1594 he must have made considerable progress in his undertaking, as in that year, Kepler tells us, Tycho Brahe was led by a Scotch correspondent to entertain hopes of the publication of the Canon or Table of Logarithms. Tycho's informant is not named, but is

* See note, p. 88.

c 2 generally

xvi Introduction.

generally believed to have been Napier's friend, Dr Craig. The computation of the Table or Canon, and the preparation of the two works explanatory of it, the Constructio and Descriptio, must, however, have occupied years. The Canon, with the description of its nature and use, made its appearance in 1614. The method of its construction, though written several years before the Descriptio, was not published till 16 19.

Napier at the same time devised several mechanical aids to computation, a description of which he published in 161 7, 'for the sake of those who may prefer to work with the natural numbers,' the most important of these aids being named Rabdologia, or calculation by means of small rods, familiarly called * Napier's bones.'

The invention of logarithms was welcomed by the greatest mathematicians, as giving once for all the long- desired relief from the labour of calculation, and by none more than by Henry Briggs, who thenceforth devoted his life to their computation and improvement. He twice visited Napier at Merchiston, in 161 5 and 1616, and was preparing again to visit him in 161 7, when he was stopped by the death of the inventor. The strain involved in the computation and perfecting of the Canon had been too great, and Napier did not long survive its completion, his death occurringon the 4th of April 161 7. He was buried near the parish church of _St Cuthbert's, outside the West Port of Edinburgh.

It has been stated that Napier dissipated his means

on

Introduction. xvii

on his mathematical pursuits. The very opposite, how- ever, was the case, as at his death he left extensive estates in the Lothians, the Lennox, Menteith, and else- where, besides personal property which amounted to a large sum.

For fuller information regarding John Napier, the reader is referred to the Memoirs, published by Mark Napier in 1834, from which the above particulars are. mainly derived.

The * Mirifici Logarithmorum Canonis Constructio' is the most important of all Napier's works, presenting as it does in a most clear and simple way the original con- ception of logarithms. It is, however, so rare as to be very little known, many writers on the subject never having seen a copy, and describing its contents from hearsay, as appears to be the case with Baron Maseres in his well-known work, * Scriptores Logarithmici,' which occupies six large quarto volumes.

In view of such facts the present translation was undertaken, which, it is hoped, will be found faithfully to reproduce the original. In its preparation valuable assistance was received from Mr John Holliday and Mr A. M. Laughton. The printing and form of the book follow the original edition of 16 19 as closely as a transla- tion will allow, and the head and tail pieces are in exact facsimile. To the work are added a few explanatory notes.

The second part of the volume consists of a Catalogue

c Z of

xviii Introduction.

of the various editions of Napier's works, giving title- page, full collation, and notes, with the. names of the principal public libraries in the country, as well as of some on the Continent, which possess copies. No simi- lar catalogue has been attempted hitherto, and it is believed it will prove of considerable interest, as show- ing the diffusion of Napier's writings in his own time, and their location and comparative rarity now. Ap- pended are notes of a few works by other authors, which are of interest in connection with Napier's writings.

It will be seen from the Catalogue that Napier's theo- logical work went through numerous editions in English, Dutch, French, and German, a proof of its widespread popularity with the Reformed Churches, both in this coun- try and on the Continent. The particulars now given also show that a statement in the Edinburgh edition of 1611 has been misunderstood. Napier's reference to Dutch editions was supposed by his biographers to apply to the German translation of Wolffgang Mayer, the Dutch translation by Michiel Panneel, being appa- rently unknown to them. His arithmetical work, Rab- dologia, also seems to have been very popular. It was reprinted in Latin, and translated into Italian and Dutch, abstracts also appearing in several languages.

Rather curiously, his works of greatest scientific interest, the Descriptio and Constructio have been most neglected. The former was reprinted in 1620, and also in Scriptores Logarithmici, besides being translated

into

Introduction. xIx

into English. The latter was reprinted in 1620 only. This neglect is no doubt largely accounted for by the advantage for practical purposes of tables computed to the base 10, an advantage which Napier seems to have been aware of even before he had made public his in- vention in 1 61 4.

For the completeness of the Catalogue I am very largely indebted to the Librarians of the numerous libraries referred to. I most cordially thank them for their kind assistance, and for the very great amount of trouble they have taken to supply me with the informa- tion I was in search of. To Mr Davidson Walker my hearty thanks are also due for assistance in collating works in London libraries.

I have only to add that any communications regarding un-catalogued editions or works relating to Napier will be gladly received.

W. R. MACDONALD.

I Forres Street, Edinburgh, December 1^, 1888.

C 4

THE

CONSTRUCTION OF THE

WONDERFUL CANON OF

LOGARITHMS;

And their relations to their own natural numbers ;

WITH

An Appendix as to the making of another and better kind of Logarithms,

TO WHICH ARE ADDED

Propositions for the solution of Spherical Triangles by an easier method : with Notes on them and 07i the above-men- tioned Appendix by the learned Henry Briggs.

By the Author and Inventor, John Napier, Baron of Merchiston, &c., in Scotland.

Printed by Andrew Hart,

OF EDINBURGH;

IN THE Year of our Lord, 1619.

Translated from Latin into English by William Rae Macdonald, 1888.

p«Gp53p

TO THE READER STUDIOUS OF

THE MATHEMATICS, GREETING.

Ryeral years ago (^Reader, Lover of the Mathe- matics) my Father, of memory always to be re- vered, made public the use of the Wonderful Canon of Logarithms ; but, as he himself men- tioned on the seventh and on the last pages of the Loga- rithms, he was decidedly against committing to types the theory and method of its creation^ until he had ascertained the opinion and criticism on the Canon of those who are versed in this kind of learning.

But, since his departure from, this life, it has been made plain to me by unmistakable proofs, that the most skilled in the mathematical sciences consider this new invention of very great importance, and that nothing more agreeable to theTn could happen, than if the construction of this Won- derful Canon, or at least so much as might suffice to ex- plain it, go forth into the light for the public benefit.

Therefore, although it is very manifest to me that the Author had not put the finishing touch to this little treat- ise, yet I have done what in me lay to satisfy their most honourable request, and to afford some assistance to those especially who are weaker in such studies and are apt to stick on the very threshold.

A 2 Nor

To THE Reader.

Nor do I doubt, but that this posthumous work would have seen the light in a much more perfect and finished state, if God had granted a longer enjoyment of life to the Author, my most dearly loved father, in whom, by the opinion of the wisest men, among other illustrious gifts this showed itself pre-emine^it, that the most difficult mat- ters were unravelled by a sure and easy method, as well as in the fewest words.

You have then (kind Reader^ in this little book most a^nply unfolded the theory of the construction of logarithms, {here called by him artificial numbers, for he had this treatise written out beside him several years before the word Logarithm was invented,) in which their nature, characteristics, and variotis relations to their natural numbers, are clearly demonstrated.

It seemed desirable also to add to the theory an Appendix as to the construction of another and better kind of loga- rithms (mentioned by the Author in the preface to his Rabdologiae) in which the logarithm of unity is o.

After this follows the last fruit of his labours, pointing to the ultimate perfecting of his Logarithmic Trigonometry, namely certain very remarkable propositions for the resolu- tion of spherical triangles not quadrantal, without dividing them into quadrantal or rectangular triangles. These propositions, which are absolutely general, he had deter- mined to reduce into order and successively to prove, had he not been snatched away from us by a too hasty death.

We have also taken care to have printed some Studies on the above-mentioned Propositions, and on the new kind of Logarithms, by that most excellent Mathematician Henry Briggs, public Professor at London, who for the singular friendship which subsisted between him and my father of illustrious memory, took upon himself, in the most willing spirit, the very heavy labour of computing this new Canon, the method of its creation and the explanation of its use

being

h

To THE Reader.

being left to the Inventor, NoWy however, as he has been called away from this life, the burden of the whole business would appear to rest on the shoulders of the most learned Briggs, on whom, too, would appear by so7ne chance to have fallen the task of adorning this Sparta.

Meanwhile [Reader) enjoy the fruits of these labours such as they are, and receive them in good part according to your culture.

Farewell,

Robert Napier, Son.

A 3

THE CONSTRUCTION OF THE WO^pET{FUL CA!J\(p!J\(^

OF LOGARITHMS; (HEREIN CALLED BY THE AUTHOR

THE ARTIFICIAL TABLE )

and their relations to

their natural

numbers.

Logarithmic Table is a small table by the use ^ of which we can obtain a knowledge of all geo- ^ metrical dimensions and motions in space^ by a\ very easy calculation.

T is deservedly called very small, because it does not exceed in size a table of sines ; very easy, because by it all multiplications, divisions, and the more difficult extractions of roots are avoided ; for by only a very few most easy addi- tions, subtractions, and divisions by two, it meas- ures quite generally all figures and motions.

It is picked out from numbers progressing in continuous proportion,

A 4 2. Of

I

8 Construction of the Canon.

2. Of continuous progressions, an arithmetical is one which proceeds by equal intervals ; a geometrical, one which advances by unequal and proportionally increasing or decreasing intervals.

Arithmetical progressions : i, 2, 3, 4, 5, 6, 7, &c. ; or 2, 4, 6, 8, 10, 12, 14, 16, &c. Geometrical progressions: i, 2, 4, 8, 16, 32, 64, &c. ; or 243, Zi, 27, 9, 3, I.

3. /?^ /^^^^ progressions we require accuracy and ease in working. Accuracy is obtained by taking large numbers for a basis ; but large numbers are most easily made from small by adding cyphers, (Icro^^

Thus instead of 1 00000, which the less experi- enced make the greatest sine, the more learned put iQOOQOOO, whereby the difference of all sines is better expressed. Wherefore also we use the same for radius and for the greatest of our geo- metrical proportionals.

4. In computing tables, these large numbers may again be made still larger by placing a period after the number and adding cyphers.

Thus in commencing to compute. Instead of 1 0000000 we put 1 0000000. 0000000, lest the most minute error should become very large by fre- quent multiplication.

5. In numbers distinguished thus by a period in their 7nidst, whatever is written after the period is a fraction, the denominator of which is unity with as many cyphers after it as there are figures after the period.

Thus 10000000.04 is the same as loooooooj^ ; also 25.803 is the same as 25Yf§f ; also 9999998.

000502 1

Construction of the Canon. 9

0005021 is the same as 9999998x^-(j^§g§J, and so of others.

6. When the tables are computed, the fractions following the period may then be rejected without any sensible error. For in our large numbers, an error which does not exceed unity is insensible and as if it were none.

Thus in the completed table, instead of 9987643.8213051, which is 9987643iwM§OT» we may put 9987643 without sensible error.

7. Besides this, there is another rule for accuracy ; that is to say, when an unknown or incommensurable quantity is included between numerical limits not differing by many units.

Thus if the diameter of a circle contain 497 parts, since it is not possible to ascertain precisely of how many parts the circumference consists, the more experienced, in accordance with the views of Archimedes, have enclosed it within limits, namely 1562 and 1 56 1. Again, if the side of a square contain 1000 parts, the diagonal will be the square root of the number 2000000. Since this is an in- commensurable number, we seek for its limits by extraction of the square root, namely 141 5 the greater limit and 1414 the less limit, or more accurately 14142828 ^^e greater, and 14142!^^ the less; for as we reduce the difference of the limits we increase the accuracy. In place of the unknown quantities themselves, their

limits are to be added, subtracted, multiplied, or divided^

according as there may be need.

8. The two limits of one quantity are added to the two limits of another, when the less of the one is added to the

B less

b

a 1-

lo Construction of the Canon.

less of the other, and the greater of the one to the greater

of the other.

Thus let the line a b c h^ divided into two parts, a b and b c. Let a b lie between the limits 123.5 the greater and 123.2 the less. Also let b c lie between the limits 43.2 the greater and 43.1 the less. Then the greater being added to the greater and the less to the less, the whole line a c will lie between the limits 166.7 ^^^ 166.3.

9. The two limits of one quantity are multiplied into the two limits of another, when the less of the one is multiplied into the less of the other, and the greater of the one into the greater of the other,

Thus let one of the quantities a b lie

between the limits 10.502 the greater and 10.500 the less. And let the other a c lie between the limits 3.216 the greater and 3.215 the less. Then 10.502 being multiplied into 3.216 and 10.500 into 3.215, the limits will become

33.774432 and 33.757500, between which the area

oi a b c d will lie.

10. Subtraction of limits is performed by taking the greater limit of the less quantity from the less of the greater, and the less limit of the less quantity from the greater of the greater.

Thus, in the first figure, if from the limits of a c, which are 166.7 ^^^ 166.3, Y^^ subtract the limits of ^ c, which are 43.2 and 43.1, the limits oi a b become 123.6 and 123.1, and not 123.5 ^^<^ 123.2. For although the addition of the latter to 43.2

and

Construction of the Canon. ii

and 43.1 produced 166.7 and 166.3 (^s in 8), yet the converse does not follow ; for there may be some quantity between 166.7 ^^i^ 166.3 from which if you subtract some other which is between 43.2 and 43.1, the remainder may not lie between 123.5 and 123.2, but it is impossible for it not to lie between the limits 123.6 and 123.1.

1 1 . Division of limits is performed by dividing the greater limit of the dividend by the less of the divisor^ and the less of the dividend by the greater of the divisor.

Thus, in the preceding figure, the rectangle abed lying between the hmits 33.774432 and 33.757500 may be divided by the limits of a Cy which are 3.216 and 3.215, when there will come out 10. 505^11^ and io.496ffxt for the limits of a b, and not 10.502 and 10.500, for the same reason that we stated in the case of subtraction.

12. The vulgar fractions of the limits may be removed by adding unity to the greater limit.

Thus, instead of the preceding limits of a b, namely, 10-505 gfff and io-496fff|, we may put 10-506 and 10-496. Thus far concerning accuracy ; what follows concerns ease in working.

1 3- The construction of every aritlunetical progression is easy ; not sOy however, of every geometrical progression. This is evident, as an arithmetical progression is very easily formed by addition or subtraction ; but a geometrical progression is continued by very difficult multiplications, divisions, or extrac- tions of roots. Those geometrical progressions alone are carried on

B 2 easily

12 Construction of the Canon.

easily which arise by subtraction of an easy part of the number from the whole number,

1 4. We call easy parts of a number, any parts the denomi- nators of which are made up of unity and a number of cyphers, such parts being obtained by rejecting as many of the figures at the end of the principal number as there are cyphers in the denominator.

Thus the tenth, hundredth, thousandth, IOOOO*^ 1 00000^^, 1 000000'^, 1 0000000^^ parts are easily obtained, because the tenth part of any number is got by deleting its last figure, the hundredth its last two, the thousandth its last three figures, and so with the others, by always deleting as many of the figures at the end as there are cyphers in the denominator of the part. Thus the tenth part of 99321 is 9932, its hundredth part is 993, its thou- sandth 99, &c.

15. The half twentieth, two hundredth, and other parts denoted by the number two and cyphers, are also tolerably easily obtained ; by rejecting as many of the figures at the end of the principal nmnber as there a7^e cyphers in the denominator, and dividing the remainder by two.

Thus the 2000^^ part of the number 9973218045 is 4986609, the 20000*^ part is 498660.

« 16. Hence it follows that if from radius with seven cyphers added you subtract its lopoopoo^^ part, and from the number thence arising its lopoopoo*'' part, and so on, a hundred numbers may very easily be continued geometri- cally in the proportion subsisting between radius and the sine less than it by unity, na7nely between 1 0000000 afid 9999999 ; and this series of proportionals we name the Pirst table.

Thus

Construction of the Canon.

13

First lOpOOOOO

I

table,

.0000000

9999999.0000000

♦9999999 9999998.000000 1

.9999993

9999997.0000003

:?999997

9999996.0000006

c ^

o

§'cr a-

Thus from radius, with seven cyphers added for greater accur- acy, namely, 1 0000000. 0000000, subtract i. 0000000, you get 9999999.0000000 ; from this sub- tract .9999999, you get 9999998. 000000 1 ; and proceed in this way, as shown at the side, until you create a hundred propor- tionals, the last of which, if you have computed rightly, will be 9999900.0004950.

9999900.0004950

The Second table proceeds from radius with six cyphers added, through fifty other numbers decreasing proportion- ally in the proportion which is easiest, and as near as possible to that subsisting between the first and last num- bers of the First table.

Thus the first and last numbers of the First table are 1 0000000. 0000000 and 9999900.0004950, in which proportion it is difficult to form fifty proportional numbers. A near and at the same time an easy proportion is 1 00000 to 99999, which may be continued with suf- ficient exactness by adding six cyphers to radius and continually subtracting from each number its own loqpoo*^ part in the manner shown at the side; and this table B 3 contains

Second table.

I opoopoo.ooopoo

100.000000 9999^900.000000

* 99.999000 9999800.001000

' 99.998000

9999700.003000

99.997000

9999600.006000

14 Construction of the Canon.

9P contains, besides radius which is

-: the first, fifty other proportional

►§ numbers, the last of which, if you

g" have not erred, you will find to be

9995001.222927 9995001.222927.

[This should be 9995001.224804— see note.]

1 8. T/ie Third table consists of sixty-nine columns, and in each column a7'e placed twenty-one numbers, proceeding in the proportion which is easiest, and as near as possible to that subsisting between the first and last numbers of the Second table.

Whence its first column is very easily obtained from radius with five cyphers added, by subtracting its 2000'^ part, and so from the other numbers as they arise.

First column of

Third table. In forming this progression, as

1 0000000. 00000 the proportion between 1 0000000.

5000.00000 000000, the first of the Second table,

9995000.00000 and 9995001.222927, the last of the

4997- 50000 same, is troublesome; therefore com-

QQQ0002. soooo P^^^ ^^ twenty-one numbers in the

499500125 ^^sy proportion of loooo to 9995,

9985007.49875 which is sufficiently near to it ; the

Af^r^o cr^^^A l^s^ o^ these, if you have not erred,

499^50374 ^jji ^^ 9900473.57808.

99500 1 4. 9950 1 From these numbers, when com-

^ puted, the last figure of each may be

^^ rejected without sensible error, so

"^ that others may hereafter be more

o easily computed from them. 9900473.57808

19. The first numbers of all the columns must proceed from

radius

Construction of the Canon.

15

20,

radius with four cyphers added^ in the proportion easiest and nearest to that subsisting between the first and the last numbers of the first column.

As the first and the last numbers of the first column are 10000000.0000 and 9900473.5780, the easiest proportion very near to this is 100 to 99. Accordingly sixty-eight numbers are to be con- tinued from radius in the ratio of 100 to 99 by subtracting from each one of them its hundredth part.

In the same proportion a progression is to be made from the second number of the first column through the second numbers in all the columns, and from the third through the thirdy and from the fourth through the fourth, and from the others respectively through the others.

Thus from any number in one column, by sub- tracting its hundredth part, the number of the same rank in the following column is made, and the numbers should be placed in order as fol- lows : —

^^

Proportionals of the Third Table.

First Column.		Second Column.
10000000.0000		990opoo.oooo
9995000.0000		9895050.0000
9990002.5000 9985007.4987 9980014.9950		9890102.4750 9885157.4237 98802 14.845 1
&c., continuously to		descending to
9900473-5780		9801^468.8423 B 4

Third

i6

Construction of the Canon.

Third Column.

9801000. 9796099. 9791201. 9786305. 9781412.

Pu n> w o

O 13 CI-

0000 5000 4503

8495 6967

9? n

^hence A^th, ^th^ (S^Cj up to

(i(^th Column.

&c., up to &c., up to &c., up to &c., up to &c., up to

5048858.8900 5046334.4605 504381 1.2932 5041289.3879 5038768.7435

1

finally to

4998609.4034

9703454.1539

21. Thus, in the Third table ^ between radius and half radius, you have sixty-eight numbers interpolated, in the proportion of 100 to 99, and between each two of these you have twenty numbers interpolated in the proportion of 10000 to 9995 ; and again, in the Second table, between the first two of these, namely between loooopoo and 9995000, you have fifty numbers interpolated in the pro- portion of 1 00000 to 99999; and finally, in the First table, between the latter, you have a hundred numbers interpolated in the proportion of radius or 1 0000000 to 9999999 ; and since the difference of these is never more than unity, there is no need to divide it more minutely by interpolating means, whence these three tables, after they have been completed, will suffice for computing a Loga- rithmic table.

Hithef^to we have explained how we may most easily place in tables sines or natural numbers progressing in geometrical proportion.

22. It remains, in the Third table at least, to place beside the sines or natural numbers decreasing geometrically

their

Construction of the Canon. 17

their logarithms or artificial numbers increasing arith- metically,

I 23. To increase arithmetically is, in equal times, to be aug- mented by a quantity always the same,

^123456789 10 d

a a a

Thus from the fixed point b let a line be pro- duced indefinitely in the direction of d. Along this let the point a travel from b towards d, mov- ing according to this law, that in equal moments of time it is borne over the equal spaces ^ i, i 2, 2 3, 3 4, 4 5, &c. Then we call this increase by b ly b 2y b 2iy b \, b ^, &c., arithmetical. Again, let ^ I be represented in numbers by 10, ^ 2 by 20, ^ 3 by 30, bj\ by 40, b^ 5 by 50; then 10, 20, 30, 40, 50, &c., increase arithmetically, because we see they are always increased by an equal number in equal times.

24. To decrease geometrically is this, that in equal times y first the whole quantity then each of its successive remain- ders is diminishedy always by a like proportional part,

T I 2 3 4 5 6 c:

-s-cTcr'-'-'

Thus let the line T S be radius. Along this let the point G travel in the direction of S, so that in equal times it is borne from T to i, which for example may be the tenth part of T S ; and from I to 2, the tenth part of i S ; and from 2 to 3, the tenth part of 2 S ; and from 3 to 4, the tenth part of 3 S, and so on. Then the sines T S, i S, 2 S,

C 3S,

i8 Construction of the Canon.

3 S, 4 S, &c., are said to decrease geometrically, because In equal times they are diminished by unequal spaces similarly proportioned. Let the sine T S be represented in numbers by looooooo, I S by 9000000, 2 S by 8100000, 3 S by 7290000,

4 S by 6561000; then these numbers are said to decrease geometrically, being diminished in equal times by a like proportion.

25. Whence a geometrically moving point approaching a fixed one has its velocities proportionate to its distances from tlu fixed one.

Thus, referring to the preceding figure, I say that when the geometrically moving point G is at T, Its velocity is as the distance T S, and when G is at I Its velocity is as i S, and when at 2 Its velocity Is as 2 S, and so of the others. Hence, whatever be the proportion of the distances T S, I S, 2 S, 3 S, 4 S, &c., to each other, that of the velocities of G at the points T, i, 2, 3, 4, &c., to one another, will be the same.

For we observe that a moving point is declared more or less swift, according as It is seen to be borne over a greater or less space In equal times. Hence the ratio of the spaces traversed Is neces- sarily the same as that of the velocities. But the ratio of the spaces traversed in equal times, T i, I 2, 2 3, 3 4, 4 5, &c., is that of the distances T S, I S, 2 S, 3 S, 4 S, &c.['"] Hence It follows that the ratio to one another of the distances of G from S, namely T S, i S, 2 S, 3 S, 4 S, &c., Is the same as that of the velocities of G at the points T, 1,2, 3, 4, &c., respectively.

[*] It is evident that the ratio of the spaces traversed T I, I 2, 2 3, 3 4, 4 5, &c., is that of the distances T S,

Construction of the Canon. 19

I S, 2 S, 3 S, 4 S, &c., for when quantities are con- tinued proportionally, their differences are also con- tinued in the same proportion. Now the distances are by hypothesis continued proportionally, and the spaces traversed are their differences, wherefore it is proved that the spaces traversed are continued in the same ratio as the distances.

26. The logarithm of a given sine is that number which has increased arithmetically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine.

T d S

1

g g

b c 1

1

Let the line T S be radius, and d S a given sine in the same line ; let g move geometrically from T to d in certain determinate moments of time. Again, let b i be another line, infinite to- wards i, along which, from b, let a move arithmet- ically with the same velocity as g had at first when at T ; and from the fixed point b in the direction of i let a advance in just the same moments of time up to the point c. The number measuring the line b c is called the logarithm of the given sine d S.

27. Whence nothing is the logarithm of radius.

For, referring to the figure, when g is at T making its distance from S radius, the arithmetical point d beginning at b has never proceeded thence. Whence by the definition of distance nothing will be the logarithm of radius.

C 2 28. Whence

20 Construction of the Canon.

28. Whence also it follows that the logarithm of any given sine is greater than the difference between radius and the given sine, and less than the difference between radius and the quantity which exceeds it in the ratio of radius to the given sine. And these differences are therefore called the limits of the logarithm,

o T d S

1 1

g g g

Thus, the preceding figure being repeated, and S T being produced beyond T to o, so that o S is to T S as T S to d S. I say that b c, the loga- rithm of the sine d S, is greater than T d and less than o T. For in the same time that g is borne from o to T, g is borne from T to d, because (by 24) o T is such a part of o S as T d is of T S, and in the same time (by the definition of a loga- rithm) is a borne from b to c ; so that o T, T d, and b c are distances traversed in equal times. But since g when moving between T and o is swifter than at T, and between T and d slower, but at T is equally swift with a (by 26) ; it follows that o T the distance traversed by g moving swiftly is greater, and T d the distance traversed by g moving slowly is less, than b c the distance traversed by the point a with its medium motion, in just the same moments of time ; the latter is, consequently, a certain mean between the two former.

Therefore o T is called the greater limit, and

Td

Construction of the Canon. 21

T d the less limit of the logarithm which b c represents.

9. Therefore to find the limits of the logarithm of a given

2

sine.

By the preceding it is proved that the given sine being subtracted from radius the less limit remains, and that radius being multiplied into the less limit and the product divided by the given sine, the greater limit is produced, as in the follow- ing example.

30. Whence the first proportional of the First table, which is 9999999, has its logarithm between the limits i. 000000 1 and 1. 0000000.

For (by 29) subtract 9999999 from radius with cyphers added, there will remain unity with its own cyphers for the less limit ; this unity with cyphers being multiplied into radius, divide by 9999999 and there will result i. 000000 1 for the greater limit, or if you require greater accuracy i. 000000 1 000000 1.

3 1 . The limits themselves differing insensibly, they or any- thing between them may be taken as the true logarithm.

Thus in the above example, the logarithm of the sine 9999999 was found to be either i. 0000000 or 1. 000000 10, or best of all 1.00000005. For since the limits themselves, i. 0000000 and 1. 000000 1, differ from each other by an insensible fraction like iooqqooo> therefore they and what- ever is between them will differ still less from the true logarithm lying between these limits, and by a much more insensible error.

32. There being any number of sines decreasing from radius in geometrical proportion, of one of which the logarithm or its limits is given, to find tliose of the others,

C 3 This

2 2 Construction of the Canon.

This necessarily follows from the definitions of arithmetical increase, of geometrical decrease, and of a logarithm. For by these definitions, as the sines decrease continually in geometrical propor- tion, so at the same time their logarithms increase by equal additions in continuous arithmetical pro- gression. Wherefore to any sine in the decreasing geometrical progression there corresponds a loga- rithm in the increasing arithmetical progression, namely the first to the first, and the second to the second, and so on.

So that, if the first logarithm corresponding to the first sine after radius be given, the second logarithm will be double of it, the third triple, and so of the others ; until the logarithms of all the sines be known, as the following example will show.

33- Hence the logarithms of all the proportional sines of the First table may be included between near li7nits, and conse- quently given with sufficient exactness.

Thus since (by 27) the logarithm of radius is o, and (by 30) the logarithm of 9999999, the first sine after radius in the First table, lies between the limits 1. 000000 1 and rooooooo ; necessarily the logarithm of 9999998.0000001, the second sine after radius, will be contained between the double of these limits, namely between 2.0000002 and 2.0000000 ; and the logarithm of 9999997.0000003, the third will be between the triple of the same, namely between 3.0000003 and 3.0000000. And so with the others, always by equally increasing the limits by the limits of the first, until you have completed the limits of the logarithms of all the proportionals of the First table. You may in this

way

Construction of the Canon. 23

way, if you please, continue the logarithms them- selves in an exactly similar progression with little and insensible error ; in which case the logarithm of radius will be o, the logarithm of the first sine after radius (by 31) will be 1.00000005, of the second 2.00000010, of the third 3.00000015, and so of the rest.

34. The difference of the logarithms of radius and a given sine is the logarithm of the given sine itself

This is evident, for (by 27) the logarithm of radius is nothing, and when nothing is subtracted from the logarithm of a given sine, the logarithm of the given sine necessarily remains entire.

35. The difference of the logarith7ns of two sines must be added to the logarithm of the greater that you may have the logarithm of the less, and subtracted from the loga- rithm of the less that you may have the logarithm of the greater.

Necessarily this is so, since the logarithms in- crease as the sines decrease, and the less loga- rithm is the logarithm of the greater sine, and the greater logarithm of the less sine. And therefore it is right to add the difference to the less loga- rithm, that you may have the greater logarithm though corresponding to the less sine, and on the other hand to subtract the difference from the greater logarithm that you may have the less logarithm though corresponding to the greater sine.

36. The logarithms of similarly proportioned sines are equi- different.

This necessarily follows from the definitions of a logarithm and of the two motions. For since by

C 4 these

24 Construction of the Canon.

these definitions arithmetical increase always the same corresponds to geometrical decrease similarly- proportioned, of necessity we conclude that equi- different logarithms and their limits correspond to similarly proportioned sines. As in the. above example from the First table, since there is a like proportion between 9999999.0000000 the first proportional after radius, and 9999997.0000003 the third, to that which is between 9999996.0000006 the fourth and 9999994.0000015 the sixth ; there- fore 1.00000005 the logarithm of the first differs from 3.00000015 the logarithm of the third, by the same difference that 4.00000020 the logarithm of the fourth, differs from 6.00000030 the logarithm of the sixth proportional. Also there is the same ratio of equality between the differences of the respective limits of the logarithms, namely as the differences of the less among themselves, so also of the greater among themselves, of which loga- rithms the sines are similarly proportioned.

37- Of three sines continued in geometrical proportion, as the square of the mean equals the product of the extremes, so of their logarithms the double of the mean equals the sum of the extremes. Whence any two of these logarithms being given, the third becomes known.

Of the three sines, since the ratio between the first and the second is that between the second and the third, therefore (by 36), of their logarithms, the difference between the first and the second is that between the second and the third. For example, let the first logarithm be represented by the line b c, the second by the line b d, the third by the line b e, all placed in the one line b c d e, thus : —

and

Construction of the Canon. 25

and let the differences c d and d e be equal. Let b d, the mean of them, be doubled by producing the line from b beyond e to f, so that b f is double b d. Then b f is equal to both the lines b c of the first logarithm and b e of the third, for from the equals b d and d f take away the equals c d and d e, namely c d from b d and d e from d f, and there will remain b c and e f necessarily equal. Thus since the whole b f is equal to both b e and e f, therefore also it will be equal to both b e and b c, which was to be proved. Whence follows the rule, if of three logarithms you double the given mean, and from this subtract a given extreme, the remaining extreme sought for becomes known ; and if you add the given extremes and divide the sum by two, the mean becomes known.

38. Of four geometrical proportionals, as the product of the means is equal to the product of the extremes ; so of their logarithms i the sum of the means is equal to the sum of the extremes. Whence any three of these logarithms being give7iy the fourth becomes known.

Of the four proportionals, since the ratio be- tween the first and second is that between the third and fourth ; therefore of their logarithms (by 36), the difference between the first and second is that between the third and fourth. Hence let such quantities be taken in the line b f as that b a

b a c d e g f

may represent the first logarithm, b c the second, b e the third, and b g the fourth, making the dif-

D ferences

26 Construction of the Canon.

ferences a c and e g equal, so that d placed in the middle of c e is of necessity also placed in the middle of a g. Then the sum of b c the second and b e the third is equal to the sum of b a the first and b g the fourth. For (by 37) the double of b d, which is b f, is equal to b c and b e together, because their differences from b d, namely c d and d e, are equal ; for the same reason the same b f is also equal to b a and b g together, because their differences from b d, namely a d and d g, are also equal. Since, therefore, both the sum of b a and b g and the sum of b c and b e are equal to the double of b d, which is b f, therefore also they are equal to each other, which was to be proved. Whence follows the rule, of these four logarithms if you subtract a known mean from the sum of the known extremes, there is left the mean sought for; and if you subtract a known extreme from the sum of the known means, there is left the extreme sought for.

39. T/ie difference of the logarithms of two sines lies between two limits ; the greater limit being to radius as the differ- ence of tJie sines to the less sine, and the less limit being to radius as the difference of the sines to the greater sine,

V T c d e S

, 1 1 1 1 ,

Let T S be radius, d S the greater of two given sines, and e S the less. Beyond S T let the dis- tance T V be marked off by the point V, so that S T is to T V as e S, the less sine, is to d e, the difference of the sines. Again, on the other side of T, towards S, let the distance T c be marked off by the point c, so that T S is to T c as d S, the greater sine, is to d e, the difference of the

sines

Construction of the Canon. 27

sines. Then the difference of the logarithms of the sines d S and e S lies between the limits V T the greater and T c the less. For by hypothesis, e S is to d e as T S to T V, and d S is to d e as T S to T c ; therefore, from the nature of propor- tionals, two conclusions follow : —

Firstly, that V S is to T S as T S to c S.

Secondly, that the ratio of T S to c S is the same as that of d S to e S. And therefore (by 36) the difference of the logarithms of the sines d S and e S is equal to the difference of the loga- rithms of the radius T S and the sine c S. But (by 34) this difference is the logarithm of the sine c S itself; and (by 28) this logarithm is included between the limits V T the greater and T c the less, because by the first conclusion above stated, V S greater than radius is to T S radius as T S is to c S. Whence, necessarily, the difference of the logarithms of the sines d S and e S lies be- tween the limits V T the greater and T c the less, which was to be proved.

40. To find the limits of the difference of the logarithms of two given sines.

Since (by 39) the less sine is to the difference of the sines as radius to the greater limit of the difference of the logarithms ; and the greater sine is to the difference of the sines as radius to the less limit of the difference of the logarithms ; it follows, from the nature of proportionals, that radius being multiplied by the difference of the given sines and the product being divided by the less sine, the greater limit will be produced ; and the product being divided by the greater sine, the less limit will be produced.

D 2 Example.

28 Construction of the Canon.

Example.

THUS, let the greater of the given sines be 9999975.5000000, and the less 9999975. 0000300, the difference of these .4999700 being multiplied into radius (cyphers to the eighth place after the point being first added to both for the purpose of demonstration, although otherwise seven are sufficient), if you divide the product by the greater sine, namely 9999975.5000000, there will come out for the less limit .49997122, with eight figures after the point ; again, if you divide the product by the less sine, namely 9999975- 0000300, there will come out for the greater limit .49997124; and, as already proved, the difference of the logarithms of the given sines lies between these. But since the extension of these fractions to the eighth figure beyond the point is greater accuracy than is required, especially as only seven figures are placed after the point in the sines ; therefore, that eighth or last figure of both being deleted, then the two limits and also the difference itself of the logarithms will be denoted by the fraction .4999712 without even the smallest par- ticle of sensible error.

41. To find the logarithms of sines or natural numbers not proportionals in the First table ^ but near or between them; or at leasty to find limits to them separated by an insensible difference.

Write down the sine in the First table nearest to the given sine, whether less or greater. Seek out the limits of the table sine (by 33), and when found note them down. Then seek out the limits of the difference of the logarithms of the given

sine

Construction of the Canon. 29

sine and the table sine (by 40), either both limits or one or other of them, since they are almost equal, as is evident from the above ex- ample. Now these, or either of them, being found, add to them the limits above noted down, or else subtract (by 8, 10, and 35), according as the given sine is less or greater than the table sine. The numbers thence produced will be near limits be- tween which is included the logarithm of the given sine.

Example.

Let the given sine be 9999975.5000000, to which the nearest sine in the table is 9999975. 0000300, less than the given sine. By 33 the limits of the logarithm of the latter are 25.0000025 and 25.0000000. Again (by 40), the difference of the logarithms of the given sine and the table sine is .4999712. By 35, subtract this from the above limits, which are the limits of the less sine, and there will come out 24.5000313 and 24.5000288, the required limits of the logarithm of the given sine 9999975.5000000. Accordingly the actual logarithm of the sine may be placed without sensible error in either of the limits, or best of all (by 31) in 24.5000300.

Another Example.

LET the given sine be 9999900.0000000, the table sine nearest it 9999900.0004950. By 33 the limits of the logarithm of the latter are 1 00.0000 1 00 and 100.0000000. Then (by 40) the difference of the logarithms of the sines will be .0004950. Add this (by 35) to the above limits and they become 100.0005050 for the greater

D 3 limit,

30 Construction of the Canon.

limit, and 100.0004950 for the less limit, between which the required logarithm of the given sine is included.

42. Hence it follows that the logarithms of all the propor- tionals in the Second table may be found with sufficient exactness, or may be included between known limits differ- ing by an insensible fraction.

Thus since the logarithm of the sine 9999900, the first proportional of the Second table, was shown in the preceding example to lie between the limits 100.0005050 and 100.0004950; neces- sarily (by 32) the logarithm of the second propor- tional will lie between the limits 200.0010100 and 200.0009900 ; and the logarithm of the third pro- portional between the limits 300.0015150 and 300.0014850, &c. And finally, the logarithm of the last sine of the Second table, namely 9995001. 222927, is included between the limits 5000. 0252500 and 5000.0247500. Now, having all these limits, you will be able (by 31) to find the actual logarithms.

43. To find the logarithms of sines or nattiral numbers not proportionals in the Second table, but near or between them ; or to include them between known limits differing by an insensible fraction.

Write down the sine in the Second table near- est the given sine, whether greater or less. By 42 find the limits of the logarithm of the table sine. Then by the rule of proportion seek for a fourth proportional, which shall be to radius as the less of the given and table sines is to the greater. This may be done in one way by multi- plying the less sine into radius and dividing the product by the greater. Or, in an easier way, by

multiplying

Construction of the Canon.

31

multiplying the difference of the sines into radius, dividing this product by the greater sine, and sub- tracting the quotient from radius.

Now since (by 36) the logarithm of the fourth proportional differs from the logarithm of radius by as much as the logarithms of the given and table sines differ from each other ; also, since (by 34) the former difference is the same as the loga- rithm of the fourth proportional itself; therefore (by 41) seek for the limits of the logarithm of the fourth proportional by aid of the First table ; when found add them to the limits of the logarithm of the table sine, or else subtract them (by 8, 10, and 35), according as the table sine is greater or less than the given sine; and there will be brought out the limits of the logarithm of the given sine.

T

Example.

HUS, let the given sine be 9995000.000000. To this the nearest sine in the Second table is 9995001.222927, and (by 42) the limits of its logarithm are 5000.0252500 and 5000.0247500. Now seek for the fourth proportional by either of the methods above described ; it will be 9999998. 7764614, and the limits of its logarithm found (by 41) from the First table will be 1.2235387 and 1.2235386. Add these limits to the former (by 8 and 35), and there will come out 5001.2487888 and 5001.2482886 as the limits of the logarithm of the given sine. Whence the number 5001.2485387, midway between them, is (by 31) taken most suitably, and with no sensible error, for the actual logarithm of the given sine 9995000.

44. Hence it follows that the logarithms of all the propor-

D 4 tionals

32 Construction of the Canon.

tionals in the first column of the Third table may be found with sufficient exactness, or may be included between known limits differing by an insensible fraction.

For, since (by 43) the logarithm of 9995000, the first proportional after radius in the first column of the Third table, is 5001.2485387 with no sensible error; therefore (by 32) the logarithm of the second proportional, namely 9990002.5000, will be 10002.4970774; and so of the others, pro- ceeding up to the last in the column, namely 9900473.57808, the logarithm of which, for a like reason, will be 100024.9707740, and its limits will be 100024.9657720 and 100024.9757760.

45. To find the logarithms of natural numbers or sines not proportionals ifi the first column of the Third table, but near or between them ; or to include them between known limits differing by an insensible fraction.

Write down the sine in the first column of the Third table nearest the given sine, whether greater or less. By 44 seek for the limits of the logarithm of the table sine. Then, by one of the methods described in 43, seek for a fourth pro- portional, which shall be to radius as the less of the given and table sines is to the greater. Hav- ing found the fourth proportional, seek (by 43) for the limits of its logarithm from the Second table. When these are found, add them to the limits of the logarithm of the table sine found above, or else subtract them (by 8, 10, and 35), and the limits of the logarithm of the given sine will be brought out.

Example.

THUS, let the given sine be 9900000. The proportional sine nearest it in the first

column

Construction of the Canon. 33

column of the Third table is 9900473.57808. Of this (by 44) the limits of the logarithm are 100024.9657720 and 100024.9757760. Then the fourth proportional will be 9999521.661 1850. Of this the limits of the logarithm, deduced from the Second table (by 43), are 478.3502290 and 478.3502812. These limits (by 8 and 35) being added to the above limits of the logarithm of the table sine, there will come out the limits 100503. 3260572 and 1 00503. 3 1 600 10, between which necessarily falls the logarithm sought for. Whence the number midway between them, which is 100503.32 1029 1, may be put without sensible error for the true logarithm of the given sine 9900000. 46. Hence it follows that the logarithms of all tJie propor- tionals of the Third table may be given with sufficient exactness.

For, as (by 45) 100503.3210291 is the logarithm of the first sine in the second column, namely 9900000 ; and since the other first sines of the remaining columns progress in the same propor- tion, necessarily (by 32 and 36) the logarithms of these increase always by the same difference 100503.32 1029 1, which is added to the logarithm last found, that the following may be made. Therefore, the first logarithms of all the columns being obtained in this way, and all the logarithms of the first column being obtained by 44, you may choose whether you prefer to build up, at one time, all the logarithms in the same column, by continuously adding 5001.2485387, the difference of the logarithms, to the last found logarithm in the column, that the next lower logarithm in the same column be made ; or whether you prefer to com-

E pute

34

Construction of the Canon.

pute, at one time, all the logarithms of the same rank, namely all the second logarithms In each of the columns, then all the third, then the fourth, and so the others, by continuously adding 100503. 32 1029 1 to the logarithm in one column, that the logarithm of the same rank in the next column be brought out. For by either method may be had the logarithms of all the proportionals in this table; the last of which is 6934250.8007528, cor- responding to the sine 4998609.4034.

47. In the Third table, beside the natural numbers, are to be written their logarithns ; so that the Third table, which after this we shall always call the Radical table, may be made complete and perfect.

This writing up of the table is to be done by arranging the columns in the number and order described (in 20 and 21), and by dividing each into two sections, the first of which should contain the geometrical proportionals we call sines and natural numbers, the second their logarithms pro- gressing arithmetically by equal intervals.

The Radical Table.

Second column.

First column.
Natural numbers.	Logarithms.
10000000.0000	.0
9995000.0000	5001.2
9990002.5000	10002.5
9985007.4987	15003.7
9980014.9950	20005.0
9? p
9900473.5780	100025.0

Natural numbers. 9900000.0000 9895050.0000 9890102.4750 9885157.4237 98802 14.845 1

•X3

9801468.8423

Logarithms. 100503.3 105504.6 II0505.8

1 1 5507. 1 120508.3

•73

200528.2

and

Construction of the Canon.

35

o	69M column.
Natural numbers.	Logarithms.
4-» u o	5048858.8900 5046334.4605 504381 1.2932 5041289.3879 5038768.7435	6834225.8 6839227.1 6844228.3 6849229.6 6854230.8
	C ^ z
	4998609.4034	6934250.8

For shortness, however, two things should be borne in mind : — First, that in these logarithms it is enough to leave one figure after the point, the remaining six being now rejected, which, however, if you had neglected at the beginning, the error arising thence by frequent multiplications in the previous tables would have grown intolerable in the third. Secondly, If the second figure after the point exceed the number four, the first figure after the point, which alone is retained, is to be increased by unity : thus for 10002.48 it is more correct to put 10002.5 than 10002.4; and for 1 000. 3 500 1 we more fitly put 1000.4 than 1000.3. Now, therefore, continue the Radical table in the manner which has been set forth.

48. The Radical table being now completed, we take the numbers for the logarithmic table from it alone.

For as the first two tables were of service in the formation of the third, so this third Radical

E 2 table

36 Construction of the Canon.

table serves for the construction of the principal Logarithmic table, with great ease and no sensible error.

49. To find most easily the logarithms of sines greater than 9996700.

This is done simply by the subtraction of the given sine from radius. For (by 29) the loga- rithm of the sine 9996700 lies between the limits 3300 and 3301 ; and these limits, since they differ from each other by unity only, cannot differ from their true logarithm by any sensible error, that is to say, by an error greater than unity. Whence 3300, the less limit, which we obtain simply by subtraction, may be taken for the true logarithm. The method is necessarily the same for all sines greater than this.

50. To find the logarithms of all sines embraced within the limits of the Radical table.

Multiply the difference of the given sine and table sine nearest it by radius. Divide the pro- duct by the easiest divisor, which may be either the given sine or the table sine nearest it, or a sine between both, however placed. By 39 there will be produced either the greater or less limit of the difference of the logarithms, or else something intermediate, no one of which will differ by a sensible error from the true difference of the logarithms on account of the nearness of the num- bers in the table. Wherefore (by 35), add the result, whatever it may be, to the logarithm of the table sine, if the given sine be less than the table sine ; if not, subtract the result from the logarithm of the table sine, and there will be produced the required logarithm of the given sine.

Example.

Construction of the Canon. 37

Example.

THUS let the given sine be 7489557, of which the logarithm is required. The table sine nearest it is 7490786.61 19. From this subtract the former with cyphers added thus, 7489557.0000, and there remains 12 29.6 11 9. This being multi- plied by radius, divide by the easiest number, which may be either 748955 7.0000 or 7490786.61 19, or still better by something between them, such as 7490000, and by a most easy division there will be produced 1640. i. Since the given sine is less than the table sine, add this to the logarithm of the table sine, namely to 28891 1 1.7, and there will result 2890751.8, which equals 289075 if. But since the principal table admits neither fractions nor anything beyond the point, we put for it 2890752, which is the required logarithm.

Another Example.

LET the given sine be 7071068.0000. The table sine nearest it will be 7070084.4434. The difference of these is 983.5566. This being multiplied by radius, you most fitly divide the product by 7071000, which lies between the given and table sines, and there comes out 1390.9. Since the given sine exceeds the table sine, let this be subtracted from the logarithm of the table sine, namely from 3467125.4, which is given in the table, and there will remain 3465734.5. Wherefore 3465735 is assigned for the required logarithm of the given sine 7071068. Thus the liberty of choosing a divisor produces wonderful facility.

E 3 51. All

38 Construction of the Canon.

51. All sines in the proportion of two to one have 6931 469. 2 2 for the difference of their logarithms.

For since the ratio of every sine to its half is the same as that of radius to 5000000, therefore (by 36) the difference of the logarithms of any sine and of Its half is the same as the difference of the logarithms of radius and of its half 5000000. But (by 34) the difference of the logarithms of radius and of the sine 5000000 is the same as the logarithm itself of the sine 5000000, and this loga- rithm (by 50) will be 6931469.22. Therefore, also, 6931469.22 will be the difference of all loga- rithms whose sines are in the proportion of two to one. Consequently the double of it, namely 13862938.44, will be the difference of all loga- rithms whose sines are In the ratio of four to one ; and the triple of it, namely 20794407.66, will be the difference of all logarithms whose sines are in the ratio of eight to one.

52. All sines in the proportion often to one have 23025842.34 for the difference of their logarithms.

For (by 50) the sine 8000000 will have for Its logarithm 2231434.68; and (by 51) the difference between the logarithms of the sine 8000000 and of its eighth part 1 000000, will be 20794407.66 ; whence by addition will be produced 23025842.34 for the logarithm of the sine 1 000000. And since radius is ten times this, all sines In the ratio of ten to one will have the same difference, 23025842.34, between their logarithms, for the reason and cause already stated (in 51) in reference to the propor- tion of two to one. And consequently the double of this logarithm, namely 46051684.68, will, as re- gards the difference of the logarithms, correspond

to

Construction of the Canon.

39

to the proportion of a hundred to one ; and the triple of the same, namely 69077527.02, will be the difference of all logarithms whose sines are in the ratio of a thousand to one ; and so of the ratio ten thousand to one, and of the others as below.

53- Whefice all sines in a ratio compounded of the ratios two to one and ten to one, have the difference of their logarithms formed from the differences 6931 469. 2 2 and 23025842.34 171 the way shown in the following

Short Table.

Given Proportions of Sines.

Two

Four

Eight

Ten

20

40

80

A hundred

200

400

800

A thousand

2000

4000

to one

Corresponding

Differences of

Logarithm.

6931469.22 13862938.44 20794407.66 23025842.34

299573IX-56 36888780.78 43820250.00 46051684.68

52983153-90 59914623.12 66846092.34 69077527.02 76008996.24 82940465.46

Given Propor of Sines.	tions one
8000 to
1 0000	J)
20000	>>
40000 80000
I 00000	M
200000	»
400000 800000
lOOOOOO	»
2000000	»
4000000 8000000	>>
lOOOOOOO	»

Corresponding

Differences of

Logarithm.

89871934.68 92103369.36 99034838.58 105966307.80 112897777.02 115129211.70 122060680.92 128992150.14 ^35923619-36 138155054-04 145086523.26 152017992.48 158949461.70 161180896.38

54- To find the logarithms of all sines which are outside the limits of the Radical table.

This is easily done by multiplying the given sine by 2, 4, 8, 10, 20, 40, 80, 100, 200, or any other proportional number you please, contained in the short table, until you obtain a number within the limits of the Radical table. By 50

E 4 find

40 Construction of the Canon.

find the logarithm of this sine now contained in the table, and then add to it the logarithmic differ- ence which the short table indicates as required by the preceding multiplication.

Example.

IT is required to find the logarithm of the sine 378064. Since this sine is outside the limits of the Radical table, let it be multiplied by some proportional number in the foregoing short table, as by 20, when it will become 7561280. As this now falls within the Radical table, seek for its logarithm (by 50) and you will obtain 2795444.9, to which add 29957311.56, the difference in the short table corresponding to the proportion of twenty to one, and you have 32752756.4. Where- fore 32752756 is the required logarithm of the given sine 378064.

55. As half radius is to the sine of half a given arc, so is the sine of the complement of t lie half arc to the sine of the whole arc.

Let a b be radius, and a b c its double, on which as diameter is described a semicircle. On this lay off the given arc a e, bisect it in d, and from e in the direc- tion of c lay off e h, the complement of d e, half the given arc. Then h c is necessarily equal to e h, since the quad- rant d e h must equal i the remaining quadrant made up of the arcs a d and h c. Draw e i perpendicular to a i c, then e i

is

Construction of the Canon. 41

is the sine of the arc a d e. Draw a e ; its half, f e, is the sine of the arc d e, the half of the arc a d e. Draw e c ; its half, e g, is the sine of the arc e h, and is therefore the sine of the comple- ment of the arc d e. Finally, make a k half the radius a b. Then as a k is to e f, so is e g to e i. For the two triangles c e a and c i e are equi- angular, since i c e or ace is common to both ; and c i e and c e a are each a right angle, the former by hypothesis, the latter because it is in the circumference and occupies a semicircle. Hence a c, the hypotenuse of the triangle c e a, is to a e, its less side, as e c, the hypotenuse of the triangle c i e, is to e i its less side. And since a c, the whole, is to a e as e c, the whole, is to e i, it follows that a b, half of a c, is to a e as e g, half of e c, is to e i. And now, finally, since a b, the whole, is to a e, the whole, as e g is to e i, we necessarily conclude that a k, half of a b, is to f e, half of a e, as e g is to e i.

« 56. Double the logarithm of an arc 0/4^ degrees is the logarithm of half radius.

Referring to the preceding figure, let the case be such that a e and e c are equal. e

In that case i will fall on b, and e i will be radius ; also e f and e g will be equal, each of them being the sine of 45 a degrees. Now (by 55) the ratio of a k, half radius, to e f, a sine of 45 degrees, is likewise the ratio of e g, also a sine of

F 45

42 Construction of the Canon.

45 degrees, to e i, now radius. Consequently (by 37) double the logarithm of the sine of 45 degrees is equal to the logarithms of the extremes, namely radius and its half. But the sum of the logarithms of both these is the logarithm of half radius only, because (by 27) the logarithm of radius is nothing. Necessarily, therefore, the double of the logarithm of an arc of 45 degrees is the logarithm of half radius.

57. The sum of the logarithms of half radius and any given arc is equal to the sum of the logarithms of half the arc and the complement of the half arc. Whence the loga- rithm of the half arc may be found if the logarithms of the other three be given.

Since (by 55) half radius is to the sine of half the given arc as the sine of the complement of that half arc is to the sine of the whole arc, there- fore (by 38) the sum of the logarithms of the two extremes, namely half radius and the whole arc, will be equal to the sum of the logarithms of the means, namely the half arc and the complement of the half arc. Whence, also (by 38), if you add the logarithm of half radius, found by 51 or 56, to the given logarithm of the whole arc, and subtract the given logarithm of the complement of the half arc, there will remain the required logarithm of the half arc.

Example.

LET there be given the logarithm of half radius (by 51) 6931469; also the arc 69 degrees 20 minutes, and its logarithm 665143. The half arc is 34 degrees 40 minutes, whose

logarithm

Construction of the Canon. 43

logarithm is required. The complement of the half arc is 55 degrees 20 minutes, and its loga- rithm 1954370 is given. Wherefore add 6931469 to 665143, making 7596612, subtract 1954370, and there remains 5642242, the required logarithm of an arc of 34 degrees 40 minutes.

58. When the logarithms of all arcs not less than 45 degrees are given^ the logarithms of all less arcs are very easily obtained.

From the logarithms of all arcs not less than 45 degrees, given by hypothesis, you can obtain (by 57) the logarithms of all the remaining arcs de- creasing down to 22 degrees 30 minutes. From these, again, may be had in like manner the loga- rithms of arcs down to 1 1 degrees 1 5 minutes. And from these the logarithms of arcs down to 5 degrees 38 minutes. And so on, successively, down to I minute.

59- To form a logarithmic table.

Prepare forty -five pages, somewhat long in shape, so that besides margins at the top and bottom, they may hold sixty lines of figures. Divide each page into twenty equal spaces by horizontal lines, so that each space may hold three lines of figures. Then divide each page into seven columns by vertical lines, double lines being ruled between the second and third columns and between the fifth and sixth, but a single line only between the others.

Next write on the first page, at the top to the left, over the first three columns, "o degrees''; and at the bottom to the right, under the last

F 2 three

44 Construction of the Canon.

three columns, "89 degrees''. On the second page, above, to the left, " 1 degree'' ; and below, to the right, **88 degrees". On the third page, above, "2 degrees"; and below, "87 degrees". Proceed thus with the other pages, so that the number written above, added to that written below, may always make up a quadrant, less i degree or 89 degrees.

Then, on each page write, at the head of the first column, ''Minutes of the degree written above" ; at the head of the second column, ''Sines of the arcs to the left " ; at the head of the third column, ^'Logarithms of the arcs to the left" ; at both the head and the foot of the third column, "Difference between the logarithms of the complementary arcs " ; at the foot of the fifth column, "Logarithms of the arcs to the right "; at the foot of the sixth column, "Sines of the arcs to the right"; and at the foot of the seventh column, "Minutes of the degree written beneath".

Then enter in the first column the numbers of minutes in ascending order from o to 60, and in the seventh column the number of minutes in descending order from 60 to o ; so that any pair of minutes placed opposite, in the first and seventh columns in the same line, may make up a whole degree or 60 minutes ; for example, enter o oppo- site to 60, I to 59, 2 to 58, and 3 to 57, placing three numbers in each of the twenty intervals between the horizontal lines. In the second column enter the values of the sines corresponding to the degree at the top and the minutes in the same line to the left ; also in the sixth column enter the values of the sines corresponding to the

degree

Construction OF the Canon. 45

degree at the bottom and the minutes in the same line to the right. Reinhold's common table of sines, or any other more exact, will supply you with these values.

Having done this, compute, by 49 and 50, the logarithms of all sines between radius and its half, and by 54, the logarithms of the other sines ; how- ever, you may, with both greater accuracy and facility, compute, by the same 49 and 50, the loga- rithms of all sines between radius and the sine of 45 degrees, and from these, by 58, you very readily obtain the logarithms of all remaining arcs less than 45 degrees. Having computed these by either method, enter in the third column the loga- rithms corresponding to the degree at the top and the minutes to the left, and to their sines in the same line at left side ; similarly enter in the fifth column the logarithm corresponding to the degree at the bottom and the minutes to the right and to their sines in the same line at right side.

Finally, to form the middle column, subtract each logarithm on the right from the logarithm on the left in the same line, and enter the difference in the same line, between both, until the whole is completed.

We have computed this Table to each minute of the quadrant, and we leave the more exact elaboration of it, as well as the emendation of the table of sines, to the learned to whom more leisure may be given.

F 3 Outline

46 Construction of the Canon.

Outline of the Construction, in another

form, of a Logarithmic Table.

60. OINCE the logarithms found fy $4. sometimes differ "^ from those found by ^% {for example, th.e logarithm of the sine 378064 is 32752756 by the former, while by the latter it is 32752741), it would seem that the table of sines is in some places faulty. Wherefore I advise the learned, who perchance may have plenty of pupils and com- putors, to publish a table of sines more reliable and with larger numbers, in which radius is made 1 00000000, that is with eight cyphers after the unit instead of seven only. Then, let the First table, like ours, contain a hundred numbers progressing in the proportion of the new radius to the sine less tlmn it by unity, namely of 1 00000000 to 99999999.

Let the Second table also contain a hundred numbers in the proportion of this new radius to the number less than it by a hundred, namely of 100000000 to 99999900.

Let the Third table, also called the Radical table, con- tain thirty-five columns with a hundred numbers in each column, and let the hundred numbers in each column pro- gress in the proportion of ten thousand to the number less than it by unity, namely of 1 00000000 to 99990000.

Let the thirty-five prop07^tionals standing first in all the columns, or occupying the second, third, or other rank, pro- gress among themselves in the proportion of 100 to 99, or of the new radius 1 00000000 to 99000000.

In continuing these proportionals and finding their loga- rithms, let the other rules we have laid down be observed.

From

Construction of the Canon. 47

From the Radical table completed in this way, you will find with great exactness (by 49 and 50) the logarithms of all sines between radius and the sine 0/4.^ degrees ; from the arc of ^^ degrees doubled, you will find {by 56) the logarithm of half radius ; having obtained all these, you will find the other logarithms by 58. Arrange cell these results as described in 59, and you will produce a Table, certainly the most excellent of all Mathe- matical tables, and pre- pared for the most important uses.

End of the Construction of the Logarithmic Table.

F 4 Appendix.

Page 48

APPENDIX.

On the Construction of another and better kind of Logarithms, namely

one in which the Logarithm of unity is o.

MoNG the various improvements ^Logarithms, the more important is that which adopts a cypher as the Logarithm of unity, and 10,000,000,000 as the Logarithm of either one tenth of unity or ten times unity. Then, these being once fixed, the Loga- rithms of all other numbers necessarily follow. But the methods of finding them are various, of which the first is as follows : —

Divide the given Logarithm of a tenth, or of ten, naTne- ly 10,000,000,000, by 5 ten times successively, and there- by the following numbers will be produced, 2000000000, 400000000, 80000000, 16000000, 3200000, 640000, 128000, 25600, 5120, 1024. Also divide the last of these by 2, ten times successively, and there will be produced ^12, 256, 128, 64, 32, 16, 8, 4, 2, I. Moreover all these num- bers are logarithms.

Thereupon let us seek for the common numbers which

correspond

Appendix. 49

corresp07id to each of them in order. Accordingly, between a tenth and unity, or between ten and unity (adding for the pwpose of calculation as many cyphers as you wish, say twelve), find four mean proportionals, or rather the least of them, by extracting the fifth root, which for ease in demonstration call A. Similarly, between A and unity, find the least of four mean proportionals, which call B. Between B and unity find four means, or the least of them, which call C. And thus proceed, by the extraction of the fifth root, dividing the interval between that last found and unity into five proportional intervals, or into four means, of all which let the fourth or least be always noted down, until you C07ne to the tenth least mean; and let them be denoted by the letters D, E, F, G, H, I, K.

When these proportionals have bee^i accurately computed, proceed also to find the mean proportional between K and unity, which call L. Then find the m.ean proportional between L and unity, which call M. Then in like manner a mean between M and unity, which call N. In the same way, by extraction of the square root, may be formed be- tween each last found number and unity, the rest of the intermediate proportionals, to be denoted by the letters O, P, Q, R,S, T,V.

To each of these proportionals in order corresponds its Logarithm in the first series. Whence i will be the Loga- rithm of the number V, whatever it may turn out to be, and 2 will be the Logarithm of the number T, and 4 of the number S, and 8 of the number R, 16 of the number Qy Z'^ of the nu7nber P, 64 of the number O, 128 of tJie number N, 256 of the number M, 512 of the number L, 1024 of the number K ; all of which is manifest from the above construction.

From these, once computed, there may then be formed both the proportionals of other Logarithms and the Loga- rithms of other proportionals,

G For

50 Appendix.

For as in statics, from weights of i, of 2, of \, of Z, and of other like numbers of pounds in the same propor- tion, every number of pounds weight, which to us now are Logarithms, may be formed by addition ; so, from the proportionals V, T, S, R, &c., which correspond to them, and from others also to be formed in duplicate ratio, the proportionals corresponding to every proposed Logarithm may be formed by corresponding multiplication of them among themselves, as experience will show.

The special difficulty of this method, however, is in finding the ten proportionals to twelve places by extraction of the fifth root from sixty places, but though this method is considerably more difficult, it is correspondingly more exact for finding both the Logarithms of proportionals and the proportionals of Logarithms. .

Another method for the easy construction

of the Logarithms of composite numbers, when the Logarithms of their primes are known.

IF two numbers with known Logarithms be multiplied together, forming a third ; the sum of their Loga- rithms will be the Logarithm of the third.

Also if one number be divided by another number, pro- ducing a third; the Logarithm of tlie second subtracted from the Logarithm of the first, leaves the Logarithm of the third.

If from a number raised to the second power, to the third power, to the fifth power, &c., certain other num- bers be produced; from the Logarithm of the first multi- plied by two, three, five, &c., the Logarithms of the others are produced.

Also

Appendix. 51

Also if from a given number there be extracted the secondy third, fifth, &c,, roots; and the Logarithm of the given number be divided by two, three, five, &c.y there will be produced the Logarithms of these roots.

Finally any common number being formed from other common numbers by multiplication, division, [raising to a power'] or extraction [of a roof] ; its Logarithm is cor- respondingly formed from their Logarithms by addition, subtraction, multiplication, by 2, 3, &c. [or division by 2, 3, &c,]: whence the only difficulty is in finding the Loga- rithms of the prime numbers ; and these may be found by the following general method.

For finding all Logarithms, it is necessary as the basis of the work that the Logarithms of some two common numbers be given or at least assumed ; thus in the fore- going first method of construction, o or a cypher was assumed as the Logarithm of the common number one, and 10,000,000,000 as the Logarithm of one-tenth or of ten. These therefore being given, the Logarithm of the number 5 (^hich is a prime number^ may be sought by the fol- lowing method. Find the mean proportional between 10 and I, namely iooowatwA^> ^^^^ l^^ arithmetical mean between 10,000,000,000 and o, namely 5,000,000,000; then find the geometrical mean between to and f^^fooooooo> namely xSBMM^iiw' ^^^^ ^^^ arithmetical mean between 10,000,000,000 and 5,000,000,000, namely 7,500,000,000;

In all continuous proportionals.

AS the sum of the means and one or other of the ex- tremes to the same extreme ; so is the difference of the extremes to the difference of the same extreme and the nearest mean,

G 2 A saving

52 Appendix.

A saving of half the Table of Loga- R I T H M s.

OF two arcs making up a quadrant, as the sine of the greater is to the sine of double its arCy so is the sine of 30 degrees to the sine of the less. Whence the Loga- rithm of the double arc being added to the Logarithm ^30 degrees, and the Logarithm of the greater being subtracted from the sum, there remains the Logarithm of the less.

The relations of Logarithms &

their natural numbers

to each other.

[A] I. T Et two sines and their Logarithms be given. If as -I — ' many numbers equal to the less sine be multiplied together as there are units in the Logarithm of the greater ; and on the other hand, as many numbers equal to the greater sine be multiplied together as there are units in the Logarithm of the less ; two equal numbers will be pro- duced, and the Logarithm of the sine so produced will be the product of the two Logarithms.

2. As the greater sine is to the less, so is the velocity of increase or decrease of the Logarithms at the less, to the velocity of increase or decrease of the Logarithms at the greater,

3. Two sines in duplicate, triplicate, quadruplicate, or other ratio, have their Logarithms in double, triple, quad- ruple, or other ratio,

4. And two sines in the ratio of one order to another order, as for instance the triplicate to the quintuplicate, or the

cube

Appendix. 53

cube to the fifth, have their Logarithms in the ratio of the indices of their orders, that is of '^ to 5.

5. If a first sine be multiplied into a second producing a third, the Logarithm of the first added to the Logarithm of the second produces the Logarithm of the third. So in division, the Logarithm of the divisor subtracted from the Logarithm of the dividend leaves the Logarithm of the quotient,

6. And if any number of equals to a first sine be multi- plied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second,

7. Any desired geometrical mean between two sines has for its Logarithm the corresponding arithmetical mean between the Logarithms of the sines.

8. If a first sine divide a third as many times successively [B] as there are units in A ; and if a second sine divides the same third as many times successively as there are units in

B ; also if the same first divide a fourth as tp^any times successively as there are units in C ; and if the same second divide the same fourth as many times successively as there are units in D : I say that the ratio of K to ^ is the same as that of C to D, and as that of the Logarithm of the second to the Logarithm of the first,

9. Hence it follows that the Logarithm of any given num- [C] ber is the number of places or figures which are contained

in the result obtained by raising the given number to the 10,000,000,000*^ power, 10. Also if the index of the power be the Logarithm of \o, the number of places^ less one, in the power or multiple^ will be the Logarithm of the root.

Suppose it is asked what number is the Loga- rithm of 2. I reply, the number of places in the result obtained by multiplying together 10,000,000,000 of the number 2.

G 3 But,

54 Appendix.

But, you will say, the number obtained by mul- tiplying together 10,000,000,000 of the number 2 is innumerable. I reply, still the number of places in it, which I seek, is numerable.

Therefore, with 2 as the given root, and 10,000,000,000 as the index, seek for the number of places in the multiple, and not for the multiple itself; and by our rule you will find 301029995 &c. to be the number of places sought, and the Loga- rithm of the number 2.

FINIS.

SOME

Page 55

SOME REMARKS

BY THE LEARNED

HENRY BRIGGS ^ On the foregoing Appendix.

The relations of Logarithms and their natural

numbers to each other, when the Logarithm

of unity is made o.

;Wo numbers with their Logarithms being given; [A] if both Logarithms be divided by some common divisor, and if each of the given numbers be multiplied by itself continuously, until the number of multiplications is exceeded, by unity only, by the quotient of the Logarithm of the other number, two equal numbers will be produced. And the Logarithm of the number produced will be the continued product of the quotients of the Logarithms and their common divisor.

Logarithms.

T 1 . 1 , (25118865 4

Let the given numbers be | 3937^^ 6

G 4. Let

56

Remarks on Appendix.

Let the common divisor be i

The first multiplied by itself 5 times J ^^^^^ 251 .88649 The second „ „ 3 ,. / 'l^555oo

Logarithms.

(o) o

(i) First power 4

(2) Second power 8

(3) Third power 12

(4) Fourth power 16 (5) Fifth power 20

251 188649 (6) Sixth power 24

Continued Proportionals.

I

25118865

63095737 158489331 39810718 I 00000000

Continued Proportionals.

I

39810718 15^933^ 630957379 251 188649

(o)

(I) (2) (3) (4)

Logarithms. O

6 12

18 24

Let the given numbers be Let the common divisor be

Another Example.

316227766

{

50118724

Logarithms.

5 7

m So„7";f' '"" "' "::" 4' "r } ■"=*- "^^zz*

Remarks on Appendix.

57

316227766 I 000000000 100 1000 316227766

(o)

(0

(2) (4)

(6) (7)

Logarith. O

5

10 20 30 35

501 18724 251 188649 630957376" 316227766

Logarith.

(0) o

(1) 7

(2) 14 (4) (5)

// should be observed that if the common divisor be unity ^ as in both the preceding examples, the product of the given Logarithms is the Logarithm of the number pro- ducedy because multiplication by unity does not increase the thing multiplied.

Third Example.

Let the given numbers be -j Let the common divisor be

343 823543

Logarithms. Quotients. 2.53529412 3 5.91568628 7

84509804

Number of Places.

3 343

6 11 7649

8 40353607

II 3841287201

18 558545864083284007

2.53529412

5.07058824

7.60588236

10.14117648

17.74705884

6 823543 (i) 5.91568628

12 678223072849 (2) II. 83137256

18 558545864083284007 (3) 17.74705884

H As

58 Remarks on Appendix.

As the quotients of the given Logarithms are 3 and 7, their product is 2 1 , which, multiplied by 84509804 the common divisor, makes 17.74705884 the Logarithm of the number produced.

It should be observed that the cube of the second number, and its equal the seventh power of the first (which some ^^// secundus solidus), contain eighteen figures, wherefore. its Logarithm has 17. in front, besides the figures follow- ing. The latter represent the Logarithm of the number denoted by the same digits, but of which 5, the first digit to the left, is alone integral, the remaining digits expressing a fraction added to the integer, thus 5iooooo§§w§ ^^• has for its Logarithm 74705884. Again, if four places remain integral, 3. must be placed in front of the Loga- rithm, thus 5585xuM§OTf ^^- ^^^ fo^ ^^^ Logarithm 3.74705884.

Hence from two given Logarithms and the si7ie of the first we shall be able to find the sine of the second.

Take some common divisor of the Logarithms, (the larger the better^ ; divide each by it. Then let the first sine multiply itself and its products continuously until the number of these products is exceeded, by unity only, by the quotient of the second Logarithm ; or until the power is produced of like name with the quotient of the second Log- arithm. The same number would be produced if tlie second sine, which is sought, were to multiply itself until it became the power of like name with the quotient of the first Logarithm, as is evident from the preceding proposition.

Therefore

Remarks on Appendix.

59

Therefore take the above power and seek for the root of it which corresponds to the quotient of the first Logarithm ; thereby you will find the required second sine. Also the Logarithm of the power itself will be the continued pro- duct of the quotients and the common divisor.

Thus let the given Logarithms be 8 and 14, and the sine corresponding to the first Logarithm be 3. A common divisor of the Logarithms is 2 ; this gives the quotients 4 and 7. If 3 multiply itself six times, you will have 2187 for the power which, in a series of continued proportionals from unity, will occupy the seventh place, and hence it may, without inconvenience, be called the seventh power. The same number, 2187, is the fourth power from unity in another series of continued proportionals, in which the first power, 6j^§gggg^, is the required second sine. The product of the quotients 4 and 7 is 28, which, multiplied by the common divisor 2, makes 56, the Logarithm of the power 2187.

Continued Proportionals.

I

3

9 27 81

243 729

2187

(o) I) (2) (3) (4) (5) (6) (7)

Logarithms. O

8

16

24

.32 40

48 56

Continued Proportionals.

I

6838521

46765372 31980598 2187

(O) (0 (2) (3) (4)

Logarithms.

14 28

42 56

It will be observed that these Logarithms differ from those employed in illustration of the previous Proposition ;

H 2 but

6o Remarks on Appendix.

but they agree in this, that in both, the Logarithm of unity is o ; and consequently the Logarithms of the same num- bers are either equal or at least proportional to each other.

[B] If a first sine divide a third, )

The first must divide the third, and the quotient of the third, and each quotient of a quotient successively as many times as possible, until the last quotient becomes less than the divisor. Then let the number of these divisions be noted, but not the value of any quotient, unless perhaps the least, to which we shall refer presently. In the same manner let the second divide the same third. And so also let the fourth be divided by each,

C first sine be 2

Thus let the ^ ^^e^f'^ " " i \ third „ „ 16

(fourth „ „ 64

The first, 2. divides the third, 16. four times; and the quotients are 8, 4, 2, i. The second, 4. divides the same third, 16. two times ; and the quotients are 4, i. There- fore A will be 4, and B will be 2,

In the same manner the first, 2. divides the fourth, 64. six times ; and the quotients are ^2, 16, 8, 4, 2, i. The second, 4. divides the fourth, 64. three times ; and the quoti- ents are 16, 4, i. Therefore C will be 6, and D will be 3.

Hence I say that, as K, a^, is to B, 2. so is C, 6. to D, 3. and so is the Logarithm of the second to the Logarithm of the first.

If in these divisions the last and smallest quotient be everywhere unity ^ as in these four cases, the numbers of

the

Remarks on Appendix. 6i

the quotients and the Logarithms of the divisors will be reciprocally proportional.

Otherwise the ratio will not be exactly the same on both sides ; nevertheless y if the divisors be very small, and the dividends sufficiently large, so that the quotients are very many, the defect from proportionality will scarcely, or not even scarcely, be perceived.

Hence it follows that the logarithm ) [C]

Let two numbers be taken, lo and 2, or any others you please. Let the Logarithm of the first, namely 100, be given ; it is required to find the Logarithm of the second. In the first place, let the second, 2. 77iultiply itself contin- uously until the number of the products is exceeded, by unity only, by the given Logarithm of the first. Then let the last product be divided as often as possible by the first number, 10. and again in like manner by the second number, 2. The number of quotients in the latter case will be 100, i^for the product is its hundredth power ; and if a number be mul- tiplied by itself a given number of times forming a certain product, then it will divide the product as many times and once more ; for example, if Z ^^ multiplied by itself four times it makes 243, and the same 3 divides 243 five times, the quotients being ^i, 27, 9, 3, i.) In the former case, where the product is continually divided by 10, it is mani- fest that the number of quotients falls short of the number of places in the dividend by one only. Therefore (by tJie preceding proposition) since the same product is divided by two given numbers as often as possible, the numbers of the quotients and the Logarithms of the divisors will be recip- rocally proportional. But, the number of quotients by the second being equal to the Logarithm of the first, the num-

H 3 ber

62 Remarks on Appendix.

ber of quotients by the first, that is the number of places in the product less one, will be equal to the Logarithm of the second.

Number of Places.
	I	0
I	2	I
I 2 3	4 16 256	2 4 8
4	1024	10
7 13 25 31	1048576 1099511627776 1208925819614 1267650600228	20 40 80 100

61 121 241 302

16069379676

25822496318

666801 3 1 608

107150835165

200

400

800

1000

603 1205 2409 301 1

114813014767 131820283599 17316587168 19950583591

2000

4000

8000

1 0000

Here we see that if we assume the Logarithm of 10 to be 10, the number of places in the tenth power is 4, wherefore the logarithm of 2 will be 3 and something over. The number of places in the hundredth power is 31 ; in the thousandth, 302 ; in the ten thousandth, 30TI ; and generally the more products we take the more nearly do we approach the true Logarithm sought for. For when the products are few, the fraction adhering to

the

Remarks on Appendix. 63

the last quotient disturbs the ratio a little; but if we assume the Logarithm of 10 to be 10,000,000,000, and if 2 be multiplied by itself continuously until the number of products is exceeded, by one only, by the given Logarithm ; then the number of places, less one, in the last product, will give the Logarithm of 2 with sufficient accuracy, because in large numbers the small fraction adhering to the last quotient will have no effect in disturbing the proportion.

THE END.

H 4 SOME

Page 64

SOME VERY REMARKABLE

PROPOSITIONS FOR THE

solution of spherical triangles

with wonderful ease.

To solve a spherical triangle without dividing it into two quadrantal or rectangular triangles.

IvEN three sides, to find any angle. And conversely y Given three angles, to find any side.

This is best done by the three methods explained in my work on Logarithms, Book II. chap, sects. 8, 9, 10.

VI

Given the side AD, & the angles 1l> & ^, to find the side A B. Multiply the sine of A D by the sine of D ; divide the pro- duct by the sine of B, and you will have the sine of A B.

4. Given

Trigonometrical Propositions. 65

4. Given the side A D, cSr' the angles X^ & V>, to find the

side B D.

Multiply radius by the sine of the complement of D; divide by the tangent of the complement of A D, and you will obtain the tangent of the arc C D : then multiply the sine of C D by the tangent of D ; divide the product by the tangent of B, and the sine of B C will result : add or subtract B C and C D, and you have B D.

5. Given the side A D, dr* the angles D dr' B, to fina the

angle A.

Multiply radius by the sine of the complement of A D ; divide by the tangent of the complement of D, and the tangent of the complement of C A D will be produced ; whence we have CAD itself Similarly multiply the sine of the complement of B by the sine of C A D ; divide by the sine of the complement of D, and the sine of B A C will be produced ; which being added to or subtracted from CAD, you will obtain the required angle BAD.

6. Given A D, df the angle D with the side B D, /<5> find

the angle B.

Multiply radius by the sine of the complement of D ; divide by the tangent of the complement of A D, and the tangent of C D will be produced ; Its arc C D subtract from, or add to, the side B D, and you have B C : then multiply the sine of C D by the tangent of D ; divide the product by the sine of B C, and you have the tangent of the angle B.

7. Given A D, <S^ the angle D with the side B D, to find

the side A B.

Multiply radius by the sine of the complement of D ; divide the product by the tangent of the com- plement of A D, and the tangent of C D will be

I produced ;

66 Trigonometrical Propositions.

produced ; its arc C D subtract from, or add to, the given side B D, and you have B C. Then multiply the sine of the complement of A D by the sine of the complement of B C ; divide the product by the sine of the complement of C D, and the sine of the complement of A B will be produced ; hence you have A B itself

Given A D, & the angle D with the side B D, to find the angle A.

This follows from the above, but the problem would require the " Rule of Three " to be applied thrice. Therefore substitute A for B and B for A, and the problem will be as follows : —

Given B D dr^ D, with the side A D, to find the angle B.

This is exactly the same as the sixth problem, and is solved by the ** Rule of Three " being applied twice only.

8. Given A D, df the angle D with the side A B, to find

the angle B.

Multiply the sine of A D by the sine of D ; divide the product by the sine of A B, and the sine of the angle B will be produced.

9. Given A D, d^ the angle D with the side A B, to find

the side B D.

Multiply radius by the sine of the complement of D, divide the product by the tangent of the comple- ment of A D, and the tangent of the arc C D will be produced. Then multiply the sine of the comple- ment of C D by the sine of the complement of A B, divide the product by the sine of the complement of A D, and you have the sine of the complement of B C. Whence the sum or the difference of the arcs B C and C D will be the required side B D.

10. Given

Trigonometrical Propositions. 67

TO. Given A T), & the angle D with the side A B, /f^ find the angle A.

Multiply radius by the sine of the complement of A D, divide the product by the tangent of the com- plement of D, and the tangent of the complement of CAD will be produced, giving us C A D. Again, multiply the tangent of A D by the sine of the com- plement of C A D, divide the product by the tan- gent of A B, and the sine of the complement of B A C will be produced, giving B A C. Then the sum or difference of the arcs B A C and CAD will be the required angle BAD.

1 1. Given A D, df the angle D with the angle A, to find the

side A B.

Multiply radius by the sine of the complement of A D, divide the product by the tangent of the com- plement of D, and you have the tangent of the com- plement of C A D ; CAD being thus known, the difference or sum of the same and the whole angle A is the angle B A C. Multiply the tangent of A D by the sine of the complement of C A D ; divide the product by the sine of the complement of B A C, and you will have the tangent of A B.

1 2. Given A D, dr' the angle D with the angle A, to find the

third angle B.

Multiply radius by the sine of the complement of A D, divide the product by the tangent of the com- plement of D, and the sine of the complement of B will be produced, from which we have the angle required.

Given A D, & the angle D with the angle A, to find the side B D.

This follows from the above, but in this form the problem would require the "Rule of Three" to be

I 2 three

68 Trigonometrical Propositions.

three times applied. Therefore substitute A for D and D for A, and the problem will be as follows : —

Given K 'D & the angle A with the angle D, to ^nd the side B A.

This is the same throughout as problem 1 1 , and is solved by applying the '' Rule of Three " twice only.

The ttse and importance of half-versed

sines.

1. /^^ IvEN two sides & the contained angle y to find the v_T third side.

From the half-versed sine of the sum of the sides subtract the half - versed sine of their difference ; multiply the remainder by the half-versed sine of the contained angle ; divide the product by radius ; to this add the half-versed sine of the difference of the sides, and you have the half-versed sine of the required base.

Given the base and the adjacent angles, the verti- cal angle will be found by similar reasoning.

2. Conversely, given the three sides, to find any angle.

From the half-versed sine of the base subtract the half- versed sine of the difference of the sides -multi- plied by radius ; divide the remainder by the half- versed sine of the sum of the sides diminished by the half-versed sine of their difference, and the half- versed sine of the vertical angle will be produced.

Given the three angles, the sides will be found by similar reasoning.

3. Given two arcs, to find a third, whose sine shall be equal to

the difference of the sines of the given arcs.

Let

Trigonometrical Propositions. 69

Let the arcs be 38° i' and yf. Their comple- ments are 51° 59^ and 13°. The half sum of the complements is 32° 29', the half difference 19° 29', and the logarithms are 621656 and 1098014 respec- tively. Adding these, you have 17 19670, from which, subtracting 693147, the logarithm of half radius, there will remain 1026523, the logarithm of 21°, or thereabout. Whence the sine of 21°, namely 358368, is equal to the difference of the sines of the arcs yy° and 38° i\ which sines are 974370 and 6 1 589 1, more or less.

Given an arc, to find the Logarithm of its versed sine, [a]

Let the arc be 13°; its half is 6° 30', of which the logarithm is 2178570. From double this, namely 4357140, subtract 693147, and there will remain 3663993. The arc corresponding to this is 1° 28', and the number put for the sine is 25595 ; but this

is also the versed sine of 13'

^

'k

5. Given two arcs, to find a third whose sine shall be equal to

the sum of the sines of the given arcs.

Let the arcs be 38° \' and 1° 28'; their sum is 39° 29' and their difference 36° 33', also the half sum is 19° 44' and the half difference 18° i6^ Wherefore add the logarithm of the half sum, viz. 1085655, to the logarithm of the difference, viz. 5 183 13, and you have 1603968; from this subtract the logarithm of the half difference, namely 11 601 77, and there will remain the logarithm 443791, to 'which correspond the arc 39° 56' and sine 641896. But this sine is equal, or nearly so, to the sum of the sines of 38° \' and 1° 28^ namely 615661 and 25595 respectively.

6. Given an arc & the Logarithm of its sine, to find the arc

whose versed sine shall be equal to the sine of the given arc.

I 3 Let

70 . Trigonometrical Propositions.

Let the arc be 39° 56', to which corresponds the logarithm 443791, the sine being unknown. To the logarithm 443791 add 693147, the logarithm of half radius, and you have 11 36938. Halve this logarithm and you have 568469. To this corresponds the arc 34° 30', which being doubled gives 69° for the arc which was sought. This is the case since the sine of 39° 56^ and the versed sine of 69° are each equal, or nearly so, to 641800.

L

[b] Of the spherical triangle A B X^, given the sides & the contained angle, to find the base,

Et the sides be 34° and 47°, and the contained angle 1 20° 24' 49". Half the contained angle is 60° 1 2^24^", and its logarithm 141 766. To the double of the latter, namely 283533, add the logarithms of the sides, namely 581260 and 312858, and the sum is II 7765 1. This sum Is the logarithm of half the difference between the versed sine of the base and the versed sine of the difference of the sides ; it is also the logarithm of the sine of the arc 17° 56', which arc we call the ** second found," for that which follows is first found.

Halve the difference of the sides, namely 13°, and you have 6° 30^ the logarithm of which is 2178570. Double the latter and you have 4357140 for the logarithm of the half-versed sine of 13°; it Is also the logarithm of the sine of the arc 0° 44', which arc we call the '' first found."

The sum of the two arcs is 1 8° 40^ the half sum 9° 20^ and their logarithms 1139241 and 1819061 respectively. Also the difference of the two arcs is 17° 12^ the half difference 8° 36', and their logarithms 1218382 and 1900221 respectively.

Now

Trigonometrical Propositions. 71

to the logarithm 12 18382, and the sum will be 3037443 ; from this sub- tract the logarithm 190022 1 and there will remain 1137222.

Now add the logarithm of the half sum, namely 1819061,

either or

to the logarithm of the complement of the half difference, namely 11 307, and the sum will be 1830368; from this sub- tract 693147 and there will remain 11 37221. Halve the latter and you have the logarithm 56861 1, to which corresponds the arc 34° 30^ and double this arc is the base required, namely 69°.

Conversely, given the three sides, to find any angle. The solution of this problem is given in my work on Loga- rithms, Book II. chap. vi. sect. 8, but partly by logarithms and partly by prosthaphceresis of arcs.

It is to be observed that in the preceding and following problems there is no need to discriminate between the dif- ferent cases, since the form, and magnitude of the several parts appear in the course of the calculation.

Another direct converse of the preceding problem follows, —

{Given the sides and the base, to find the vertical angle^

Alve the given base, namely 69°, and you have 34° 30^ the logarithm of which is 5686 11. Double the latter and you have 11 3722 2; corres- ponding to this is the arc 18° 42^ which note as the second found.

As before, take for the first found the arc 0° 44', corresponding to the logarithm 4357140.

The complements of the two arcs are 89° 16' and 71° 18'; their half sum is 80° 17', and its logarithm

I 4 14449;

H

72 Trigonometrical Propositions.

14449; their half difference is 8° 59', and its loga- rithm 1856956. Add these logarithms and you have 1 87 1405; subtract 693 147 and there remains 11 78258. The arc corresponding to this logarithm is 17° 56', which arc we call the third found.

From the logarithm of the third found, subtract the logarithms of the given sides, namely 581260 and 312858, and there remains 283533; halve this and you have 141 766 for the logarithm of the half vertical angle 60° 12' 24^'^ The whole vertical angle sought is therefore 120° 24' 49^^

N

Another rule for finding the base by prosthaphcsresis, —

\Gwen the sides and vertical angle ^ to find the dase.]

Ote the half difference between the versed sines of the sum and difference of the sides, and also the half-versed sine of the vertical angle. Look among the common sines for the values noted, and find the arcs corresponding to them in the table. Then write for the second found the half difference of the versed sines of the sum and difference of these arcs.

Also, as before, take for the first found the half- versed sine of the difference of the sides.

Add the first and second found, and you will obtain the half- versed sine of the base sought for.

Conversely — \given the sides and the base, to find the vertical angle.]

The first found will be, as before, the half-versed sine of the difference of the sides.

From the half- versed sine of the base subtract the first found and you will have the second found.

Multiply the latter by the square of radius ; divide

by

Trigonometrical Propositions. 73

by the half difference between the versed sines of the sum and difference of the sides, and you have as quotient the half- versed sine of the vertical angle sought for.

Of five parts of a spherical triangle, given the three inter- [c] mediate, to find tlie two extremes by a single operation. Or otherwise, given the base and adjacent angles, to find the two sides,

(*) C\ F the angles at the base, write down the sum,

v^ half sum, difference and half difference, along with their logarithms.

Add together the logarithm of the half sum, the logarithm of the difference, and the logarithm of the tangent of half the base ; subtract the logarithm of the sum and the logarithm of the half difference, and you will have the first found.

Then to the logarithm of the half difference add the logarithm of the tangent of half the base ; sub- tract the logarithm of the half sum, and you will have the second found.

Look for the first and second found among the logarithms of tangents, since they are such, then add their arcs and you will have the greater side ; again subtract the less arc from the greater and you will have the less side.

A'

AnotJter way of finding the sides,

Dd together the logarithm of the half sum of the angles at the base, the logarithm of the com- plement of the half difference, and the logarithm of the tangent of half the base ; subtract the logarithm of the sum and the logarithm of half radius, and you will have the first found.

K Again,

74 Trigonometrical Propositions.

Again, add together the logarithm of the half dif- ference, the logarithm of the complement of the half sum, and the logarithm of the tangent of half the base; subtract the logarithm of the sum and the logarithm of half radius, and you will have the second found.

Proceed as above with the first and second found, and you will obtain the sides.

M

Another way of the same,

Ultiply the secant of the complement of the sum of the angles at the base by the tangent of half the base.

Multiply the product by the sine of the greater angle at the base, and you will have the first found.

Multiply the same product by the sine of the less angle, and you will have the second found, [d] Then divide the sum of the first and second found

by the square of radius, and you will have the tan- gent of half the sum of the sides.

Also subtract the less from the greater and you will have the tangent of half the difference of the sides.

Whence add the arcs corresponding to these two tangents, and the greater side will be obtained ; sub- tract the less arc from the greater and you have the less side.

Of the five consecutive parts of a spherical triangle^ given the three intermediate, to find both extremes by one oper- ation and without the need of discriminating between the several cases,

(^) ^^^ ^^ angles at the base, the sine of the half

^^ difference is to the sine of the half sum, as the sine of the difference is to a fourth which is the sum of the sines.

And

Trigonometrical Propositions. 75

And the sine of the sum is to the sum of the sines as the tangent of half the base is to the tangent of half the sum of the sides.

Whence the sine of the half sum is to the sine of the half difference of the angles as the tangent of half the base is to the tangent of half the difference of the sides.

Add the arcs of these known tangents, taking them from the table of tangents, and you will have the greater side ; in like manner subtract the less from the greater and the less side will be obtained.

FINIS.

K 2 ' SOME

Page 76

SOME NOTES

BY THE LEARNED HENRY BRIGGS

ON THE FOREGOING PROPOSITIONS.

[a] ^;^^ IvEN an arc, to find the logarithm of its versed sine. To the end of this proposition %* / should like to add

the following : —

Conversely, given the logarithm of a versed sine, to find its arc.

Add the known logarithm of the required versed sine to the loga- rithm of '^0°, viz., 693147, and half the sum will be the logarithm of half the arc sought for.

Thus let 35791 be the given logarithm of an unknown versed sine, whose arc is also unknown.

To this logarithm add 693147, and the sum will be 728938, half of which, 364469, is the logarithm of 43° 59^ 33^'- The arc of the given logarithm is therefore Zf 59' 6'^ and its versed sine is 9648389.

Again, let a negative logarithm, say —54321, be the known logarithm of the required versed sine. To this

logarithm

Notes on Trigonometrical Propositions. 77

logarithm add, as before, 693147, and the sum, that is the number remaining since the sines are contrary, will be 638826, half of which, 31 941 3, is the logarithm of 46° 36' o''. The arc of the given logarithm is therefore 93° 12' o", the versed sine of which is 105582 16, and since this is greater than radius it has a negative logarithm, namely —54321.

^d

Demonstra- tion.

, I versed sine of arc] ,

X c '^ cont. c g > pro- c h ) port.

^x c, sine of 30° o' \ cont. c g, sine of \ arc c d > pro- c b, double of line c h ) port.

Later on I observed that the sixth proposition might be proved in an exactly similar way.

Of the spherical triangle A B D ]

In finding the base we may pursue another method, namely : —

Add the logarithm of the versed sine of t lie giveyi angle to the logarithms of the given sides y and tlte stim will be the logarithm of the difference between the versed sine of the difference of tlie sides and the versed sine of tJie base required. This difference being cmisequently known, add to it the versed siyie of the difference of the sides y and the sum will be tlte versed sine of tlie base required.

For example, let the sides be 34° and 47°, their loga-

K 3 rithms

yS Notes on Trigonometrical Propositions.

rithms 581 261 and 312858, and the logarithm of the versed sine of the given angle —409615. The sum of these three logarithms is 484504, which is the logarithm of the difference between the versed sine of the base and - the versed sine of the difference of the sides.

Now the line corresponding to this logarithm, whether a versed sine or a common sine, is 6160057, and conse- quently this is the difference between the versed sine of the base and the versed sine of the difference of the sides. If to this you add the versed sine of the differ- ence of the sides, that is 256300, the sum will be the versed sine of the base required, namely 6416357, and this subtracted from radius leaves the sine of the com- plement of the base, namely 3583643, which is the sine of 21°. Consequently the base required is 69°.

Conversely, given three sides, to find any angle.

If from the logarithm of the difference between the versed sine of the base and the versed sine of the difference of the sides you sub- tract the logarithms of the sides, the remainder will be the loga- rithm of tJie versed sine of the angle sought for.

As in the previous example, let the logarithms of the sides be 581 261 and 312858. Subtract their sum, 8941 19, from the logarithm 484504, and the remainder will be the negative logarithm— 409615, which gives the versed sine of the required angle 1 20° 24' 49''.

[c] Of five parts of a spherical triangle ]

This proposition appears to be identical with the one which is inserted at the end, and distinguished like the former by (*). The latter proposition I consider much the superior. There are^ how- ever, three operations in it, the first two of which I throw into one, as they are better combined. Thus : —

Let

Notes on Trigonometrical Propositions. 79

Let there be given the base 69°, the angles at the base \^ o 2/^^,f

4" sum.

73° 36'

36° 48' 2j half sum,

53° 11' 58" complement of ^ sum.

11° 23^54" difference.

5° 41' 57" half difference. 84° 18' 3"compl.of|diff

{ Sine half difference Proper- J Sine half sum tion I. J Sine difference I Sum of sines

5° 41' 57" 36° 48' 2" 11° 23' 54"

Logarithms.

23095560

5124410

16213641

-1757509

	/ Sine of sum 73° 36'	4"	415312
Propor-	) Sum of sines		—1757509
tion 2.	\ Tangent half base 34° 30'	0"	3750122
	' Tangent ^ sum of sides 40° 30'	0"	1577301

/ Sine ^ sum of angles 36° 48' 2

Propor- I Sine ^ diff. of angles . 5° 41' 57"

tion 3. J Tangent ^ base . . 34° 30' o''

' Tangent ^ diff. of sides 6° 30' o"

5124410 23095560

3750122 21721272

40° 30' 6° 30'

34

'}

sides.

TAese are the operations described by the Author, But

I replace the first two by another ^ retaining

the third,

K 4 Proportion.

8o Notes on Trigonometrical Propositions.

Logarithms.

/ Sine compl.^ sum of angles 53°ii'58" 2222368

Proper- I Sine comply difif. of angles 84^18' 3" 49553

tion. ^ Tangent ^ base . . 34° 30' o" 3750122

( Tangent ^ sum of sides . 40° 30' o" 1 577307

Another Example. Let there be given the angle 47°, the sides containing it -j ^% ^5/ ^ \

90° 41' 16" sum.

45° 20' 38'' half sum.

44° 39' 22" compl. of half sum.

28° 29' 6" difference. 14° 14' 33'' half difference. 75° 45^ 27" compl. of half diff.

Logarithms.

/Sine comply sum of sides 44° 39^ 22" 35 26 11 8

Propor- J Sine compl. -| diff. of sides 75°45^27" 312192

tion I. \ Tan. compl. -^ vert, angle 66° 30' o" —8328403

\ Tan.^sumofangs.atbase 72° 30^ o'' — 1 1452329

/ Sine ^ sum of sides . 45° 20' 38"

Propor- 1 Sine ^ diff. of sides . 14° 14' 33"

tion 2. J Tan. compl. ^ vert, angle 66° 30' o"

( Tan. ^diff. of angs. at base 38° 30' o"

72° 30' 38° 30'

3406418

14023154

—8328403

2288333

iir

34^

C' } ^"^'

es at the base.

And these relations are all uniformly maintained,

whether

Notes on Trigonometrical Propositions. 8i

whether there be given two angles with the interjacent side or two sides with the contained angle. In each operation the important point is what occupies the third place in the proportion. In the former it is the tangent of half the base, in the latter the tangent of the comple- ment of half the vertical angle. In these examples, if the tangent or the sum of the sines be greater than radius, the logarithm is negative and has a dash preceding, for example — 8328403.

Another way of the same ]

Then divide the sum of the first and second found by [d] the square of radius, and you will have ")

To make the sense cleareVy I should prefer to write this as follows : —

Then divide both the first and second found by the square of radius, add the quotients, and you will have the tangent, &c.

This proposition is absolutely true, as well as the one preceding ; but while the former may most conveniently be solved by logarithm's y the latter will not admit of the use of logarithms throughout^ as the quotients must be added and subtracted to find the tangents ; for the utility of Logarithms is seen in proportionals ^ and there- fore in multiplication and division^ and not in addition or subtraction,

THE END. L

Notes

BY

THE TRANSLATOli

L 2

J(li:(^^^fl!AJ^C^<ks^

Notes.

Spelling of the Author's Name.

The spelling in ordinary use at the present time is Napier. The older spellings are various — for example, Napeir, Nepair, Nepeir, Neper, Nepper, Naper, Napare, Naipper. Several of these spellings are known to have been used by our author.

I adopt the modem spelling, which is that used by his biographers, and also in the 1645 edition of * A Plaine Discovery.'

If, however, the claim of present usage be set aside, a strong case might be made out for Napeir, as this was the spelling adopted in ' A Plaine Discovery,' the only book published by Napier in English. In this work is a letter signed "John Napeir" dedicating the book to James VI., and as this letter is a solemn address to the King, we may infer that the signature would be in the most approved form. The work was first issued in 1593, and the same spelling was retained in the subsequent editions during the author's lifetime, as well as in the French editions which were revised by him. In the 1645 edition, as mentioned above, the modern spelling was introduced.

The form Nepair is used in Wright's translation of the Descriptio, published in 16 16, but too much stress must not be laid on this, as very slight importance was attached to the spelling of names; thus although Briggs contributed a preface, his name is spelt in three differ- ent ways, — Brigs, Brigges, and Briggs.

In the works published in Latin the form Neperus is invariably used.

On

Notes. 85

On some Terms made use of in the Original Work.

Napier's Canon or Table of Logarithms does not contain the logarithms of equidifferent numbers, but of sines of equidifferent arcs for every minute in the quadrant. A specimen page of the Table is given in the Catalogue under the 16 14 edition of the Descriptio.

The sine of the Quadrant or Radius, which he calls Sinus Totus^ was assumed to have the value 1 0000000.

Numerus Artificialis^ or simply Artificialis^ is used in the body of the Constructio for Logarithm, the number corresponding to the logarithm being called Numerus Naturalis.

LogarithmuSi corresponding to which Numerus Vulgaris is used, is however employed in the title-page and headings of the Constructio, and in the Appendix and following papers. It is also used throughout the Descriptio published in 1614; and as the word was not invented till several years after the completion of the Constructio (see the second page of the Preface, line 12), the latter must have been written some years prior to 16 14.

For shortness, Napier sometimes uses the expression logarithm of an arc for the logarithm of the sine of an arc.

The Antilogarithm of an arc, meaning log. sine complement of arc, and the Differential of an arc, meaning log. tangent, of arc (see De- scriptio, Bk. I., chap, iii.), are terms used in the original, but as they have a different signification in modern mathematics, we do not use them in the translation.

Prosthaphceresis was a term in common use at the beginning of the seventeenth century, and is twice employed by Napier in the Spherical Trigonometry of the Constructio as well as in the Descriptio. The following short extract from Mr Glaisher's article on Napier, in the 'Encyclopaedia Britannica,' indicates the nature of this method of calculation.

The "new invention in Denmark" to which Anthony Wood refers as hav- ing given the hint to Napier was probably the method of calculation called prosthaphseresis (often written in Greek letters ■KpoaOatpaipeffis), which had its origin in the solution of spherical triangles. The method consists in the use of the formula sin a sin b = \ {cos {a - b)-cos {a + b)], by means of which the mul- tiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines ; and as such products occur in the solution

L 3 of

S6 Notes.

of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only. It seems to be due to Wittich of Breslau, who was assistant for a short time to Tycho Brahe ; and it was used by them in their calculations in 1582.

In the spherical trigonometry the notation used in the original is

either of the form 34 gr 24 49 or 34 : 24 : 49, but in the translation the form of notation used is always 34° 24' 49".

References to delay in publishing the Constructio, and to a new kind of Logarithms to Base lo.

The various passages from Napier's works bearing on these points are given below.

The first two are referred to by Robert Napier in the first page of the Preface, line 5. They appeared in the Descriptio, published in 1 6 14, — the first, entitled Admonitio, on p. 7 (Bk. I. chap, ii.), and the second, with the title Conclusio, on the 57 th or last page of the work (Bk. II. chap, vi.)

The third passage, entitled Admonitio, is printed on the back of the last page of the Table of Logarithms published along with the De- scriptio, but is omitted in many copies.

The fourth was inserted by Napier at p. 19 (Bk. I. chap, iv.) of Wright's translation, published in 161 6.

The last is the passage referred to in the second page of the Pre- face, line 18. It is the opening paragraph in the Dedication of *Rab- dologiae ' to Sir Alexander Seton.

/. From DESCRIPTIO, Book L Chapter II.

Note.

Up to this point we have explained the genesis and properties of logarithms, and we should here show by what calculations or method of computing they are to be had. But as we are issuing the whole Table containing the loga- rithms with their sines to every minute of the quadrant, we leave the Theory of their Construction for a more fitting time and pass on to their use. So that their use and advantages being first understood, the rest may either please the more if published hereafter or at least displease the less by being buried in

silence.

Notes. 87

silence. For I await the judgement and criticism of the learned on this before unadvisedly publishing the others and exposing them to the detraction of the

//. Fro7n DESCRIPTIO, Book II. Chapter VI. Conclusion. It has now, therefore, been sufficiently shown that there are Logarithms, what they are, and of what use they are : for by their help without the trouble of multiplication, division, or extraction of roots we have both demonstrated clearly and shown by examples in both kinds of Trigonometry that the arith- metical solution of every Geometrical question may be very readily obtained. Thus you have, as promised, the wonderful Canon of Logarithms with its very full application, and should I understand by your communications that this is likely to please the more learned of you, I may be encouraged also to publish the method of constructing the table. Meanwhile profit by this little work, and render all praise and glory to God the chief among workers and the helper of all good works.

///. From the End of the TABLE OF LOGARITHMS.

Note.

Since the calculation of this table, which ought to have been accomplished by the labour and assistance of many computors, has been completed by the strength and industry of one alone, it will not be surprising if many errors have crept into it. These, therefore, whether arising from weariness on the part of the computor or carelessness on the part of the printer, let the reader kindly pardon, for at one time weak health, at another attention to more important affairs, hindered me from devoting to them the needful care. But if I perceive that this invention is likely to find favour with the learned, I will perhaps in a short time (with God's help) give the theory and method either of improving the canon as it stands, or of computing it anew in an improved form, so that by the assistance of a greater number of computors it may ultimately appear in a more polished and accurate shape than was possible by the work of a single individual.

Nothing is perfect at birth.

THE END.

IV. From WRIGHTS TRANSLATION OF THE DESCRIPTIO, Book I. Chapter IV.

An Admonition.

Bvt because the addition and subtraction of these former numbers [logs, of ^ and its powers] may seeme somewhat painfull, I intend (if it shall please

God)

L 4

88 Notes.

God) in a second Edition, to set out such Logarithmes as shal make those numbers aboue written to fall upon decimal numbers, such as ioo,ooo,ocx), 200,cxx),ooo, 300,000,000, &c., which are easie to bee added or abated to or from any other number.

V. From the DEDICATION OF RABDOLOGIM.

Most Illustrious Sir, I have always endeavoured according to my strength and the measure of my ability to do away with the difficulty and tediousness of calculations, the irksomeness of which is wont to deter very many from the study of mathematics. With this aim before me, I undertook the publication of the Canon of Logarithms which I had worked at for a long time in former years ; this canon rejected the natural numbiers and the more difficult opera- tions performed by them, substituting others which bring out the same results by easy additions, subtractions, and divisions by two and by three. We have now also found out a better kind of logarithms, and have determined (if God grant a continuance of life and health) to make known their method of construc- tion and use ; but, owing to our bodily weakness, we leave the actual computa- tion of the new canon to others skilled in this kind of work, more particularly to that very learned scholar, my very dear friend, Henry Briggs, public Pro- fessor of Geometry in London.

Notation of Decimal Fractions.

In the actual work of computing the Canon of Logarithms, Napier would continually make use of numbers extending to a great many places, and it was then no doubt that the simple device occurred to him of using a point to separate their integral and fractional parts. It would thus appear that in the working out of his great invention of Logarithms, he was led to devise the system of notation for decimal fractions which has never been improved upon, and which enables us to use fractions with the same facility as whole numbers, thereby immensely increasing the power of arithmetic. A full explanation of the notation is given in sections 4, 5, and 47, but the following extract, translated from * Rabdologiae,' Bk. I. chap, iv., is interesting as being his first published reference to the subject, though the above sections from the Constructio must have been written long before that date, and the point had actually been made use of in the Canon of Logarithms printed at the end of Wright's translation of the Descriptio in 16 16.

From

Notes. 89

From RABDOLOGI^, Book I. Chapter IV.

Note on Decimal Arithmetic.

But if these fractions be unsatisfactory which have different denominators, owing to the difficulty of working with them, and those give more satisfaction whose denominators are always tenths, hundredths, thousandths, &c., which fractions that learned mathematician, ''6'm^w Sievin, in his Decimal Arithmetic denotes thus — (T), (T), (^, naming them firsts, seconds, thirds : since there is the same facility in working with these fractions as with whole numbers, you will be able after com- pleting the ordinary division, and adding a period or comma, as in the margin, to add to the dividend or to the remainder

The preceding example :— divi- sion of 861094 by 432- 118 141 402 429

86io94(i993jJI 432

1296

7" one C5^her to obtain tenths, two for hundredths, three for

I 36 thousandths, or more afterwards as required : and with these

118 000 y°^ ^'^^^ ^^ ^^^^ *° proceed with the working as above. For

141 instance, in the preceding example, here repeated, to which

^ we have added three cyphers, the quotient will become

861094,000(1993,273 1993,273, which signifies 1993 units and 273 thousandth parts

3888 / // ///

3888 or ^V^nr* or, according to Stevin, I993j2 7 3 : further the last

— ^-^1^ — remainder, 64, is neglected in this decimal arithmetic because

30 24 it is of small value, and similarly in like examples.

I 296

Simon Stevin, to whom Napier here refers, was born at Bruges in 1548, and died at The Hague in 1620. He published various mathe- matical works in Dutch. The Tract on Decimal Arithmetic, which introduced the idea of decimal fractions and a notation for them, was published in 1585 in Dutch, under the title of *De Thiende/ and in the same year in French, under the title of * La Disme.'

We find Briggs, in his ' Remarks on the Appendix,' while sometimes employing the point, also using the notation 25118865 for 2-^^^^^^^-^^ distinguishing the fractional part by retaining the line separating the numerator and denominator, but omitting the latter. The form 2I5118865 has also been used. If we take any number such as 94t¥o^^j the following will give an idea of some of the different notations employed at various times : —

©0000

940I030O050; 941305; 9 4 i 3 05; 941305; 94I1305; 94- 1305-

M Notwithstanding

90 ' Notes.

Notwithstanding the simpHcity and elegance of the last of these, it was long after Napier's time — in fact, not till the eighteenth century — that it came into general use.

The subject is referred to by Mark Napier in the ' Memoirs,' pp. 451- 455, and by Mr Glaisher in the Report of the 1873 Meeting of the British Association, Transactions of the Sections, p. 16.

On the Occurrence of a Mistake in the Computa- tion of the Second Table ; with an Enquiry into the Accuracy of Napier's Method of Computing his Logarithms.

It is evident that a mistake must somewhere have occurred in the computation of the Second table, since the last proportional therein is given (sec. 17) as 9995001.222927, whereas on trial it will be found to be 9995001.224804.

This mistake introduced an error into the logarithms of the Radical table, as the logarithm of the first proportional in that table is deduced from the logarithm of the last proportional in the Second table by finding the limits of their difference. But these limits are obtained from the proportionals themselves, and, as shown above, one of these propor- tionals was incorrect : the limits therefore are incorrect, and conse- quently the logarithm of the first proportional in the Radical table.

We see the effect of this in the logarithm of the last proportional in the Radical table, which is given (sec. 47) as 6934250.8, whereas it should be 6934253.4, the given logarithm thus being less than the true logarithm by 2.6, or rather more than a three millionth part.

The logarithms as published in the original Canon are affected by the above mistake, and also, as mentioned in sec. 60, by the imperfection of the table of sines. It seems desirable, therefore, to enquire whether in addition any error might have been introduced by the method of computation employed.

Before entering on this enquiry, we should premise that in comparing Napier's logarithms with those to the base e~^ (which is the base re- quired by his reasoning, though the conception of a base was not for- mally known to him), it must be kept in view that in making radius

10,000,000

Notes. 91

10,000,000 he multiplied his numbers and logarithms by that amount, thereby making them integral to as many places as he intended to print. In this we follow his example, omitting, however, from the formulae the indication of this multiplication.

In sec. 30, Napier shows that the logarithm of 9999999, the first proportional after radius in the First table, lies between the limits i.ooooooiooooooio etc., and i. 000000000000000 etc. And in sec. 31, he proposes to take 1.00000005, ^^^ arithmetical mean between these limits, as a sufficiently close approximation to the true logarithm ; for, the difference of this mean from either limit being .00000005, it cannot differ from the true logarithm by more than that amount, which is the twenty millionth part of the logarithm. But there can be little doubt that Napier was able to satisfy himself that the difference would be very much less, and that his published logarithms would be unaffected.

We proceed to show the precise amount of error thus introduced into the logarithm of 9999999. If we employ the formula

substituting 1 0000000 for n, and multiplying the result by 1 0000000, as before explained, we have

1.000000050000003333333583 etc. Again, if we take the arithmetical mean of the limits, carried to a similar number of places, we have

1.000000050000005000000500 etc. The error introduced is consequently

.000000000000001666666916 etc. or about a six hundred billionth part in excess of the true logarithm. It will be observed that besides being very much less, this error is in the opposite direction from that caused by the mistake in the Second table.

We have given above the analytical expression for the true logarithm, namely, 2 + _L^ + l-^ + _L + ^.1, + etc. The corresponding expression

for the arithmetical mean is ^ + 2^ + 2^ + 2^ + 2;r6 + ^^c. The latter,

therefore, exceeds the true logarithm by ^^^ + ^ + ^^ -f etc., which

multiplied by n gives ^i-. + etc., or ^^^^^y, + etc., for the error in

Napier's logarithm. So that up to the 15 th place the logarithm

'^ 2 obtained

92 Notes.

obtained by Napier's method of computation is identical with that to the base e~\ If, however, he had used the base (l — -)", where n = looooooo, then the logarithm of 9999999, multiphed by 1 0000000, as in the other two cases, would necessarily have been unity, or 1. 000000000 etc., which would have agreed with the true logarithm to the 8th place only, and would not have left his published logarithms unaffected.

The small error found above in Napier's logarithm of 9999999 is suc- cessively multiplied on its way through the tables : thus, in the First table it is multiplied by 100, in the Second by 50, and in the Third by 20 and again by 69, or in all by 6900000 ; so that, multiplying the error in the first proportional by that amount, we should have for the error in the logarithm of the last proportional of the Radical table about .0000000115. The error, however, although continually increasing, yet retains always the same ratio to the logarithm, except for a very small disturbing element to be afterwards referred to, so that the true loga- rithm will always be very nearly equal to the logarithm found by Napier's method of computation less a six hundred billionth part.

Let us take, for example, the logarithm of 5000000 or half radius. When computed according to Napier's method, we find it comes out

693147 1.80559946464604 etc. The true logarithm to the base e~^ is

6931471.80559945309422 etc. So that the difference between the two is

.00000001 155181 etc. The six hundred billionth part of the logarithm is

.00000001155245 etc. The latter agrees very closely with the difference found above, and would have agreed to the last place given except for the small disturb- ing element referred to above, which is introduced in passing from the logarithms of one table to those of the next, or in finding the logarithm of any number not given exactly in the tables as in this case of half radius, but this element is seen to have little effect in modifying the proportionate amount of the original error.

From the above example we see that the error in the logarithm found by Napier's method amounts only to unity in the T5th place, so that his method of computation clearly gives accurate results far in excess of his requirements. But it is easy to show that Napier's method may be

adapted

Notes. 93

adapted to meet any requirements of accuracy. In sec. 60, Napier, in suggesting the construction of a table of logarithms to a greater number of places, proposes to take 1 00000000 as radius. The effect of this would be to throw still further back the error involved in taking the arithmetical mean of the limits for the true logarithm. Thus, using the formula given, substituting 1 00000000 for «, and multiplying the re- sult by that amount as already explained, we should have for the true logarithm of 99999999, the first proportional after radius in the new First table,

1.000000005000000033333 etc. If we take the arithmetical mean of the hmits, we have

1.000000005000000050000 etc. This brings out a difference of

.000000000000000016666 etc., or a sixty thousand billionth part of the logarithm. We see that the logarithms only begin to differ in the i8th place, and that thus to how- ever many places the radius is taken, the logarithms of proportionals deduced from it will be given with absolute accuracy to a very much greater number of places.

To ensure accuracy in the figures given above, the three preparatory tables were recomputed strictly according to the methods described in the Constructio, fourth proportionals being found in all the preceding tables, and both limits of their logarithms being calculated, the work being carried to the 27th place after the decimal point.

As logarithms to base e~^ are now quite superseded, it is not worth while printing these preparatory tables. The following values (pp. 94-95), however, may be of service for comparison, and as a check to any one who may desire to work out for himself the tables and examples in the Constructio. The values given are the first proportional after radius, and the last proportional in each of the three tables, and also in the Third table, the last proportional in col. i, and the first proportionals in col. 2 and 69. Opposite these are given their logarithms to base e~^ com- puted, first, according to Napier's method, and second, by the present method of series which gives the value true to the last place, which is increased by unit when the next figure is 5 or more. The propor- tionals and logarithms are each multiplied by 1 0000000, as explained above.

Though the logarithms in the Canon of 16 14 were affected by the

M 3 mistake

94

Notes.

	Proportionals.
First Table.
First proportional after radius,	9999999.
The last proportional.	9999900.00049499838300392 1 2 1 747 1
Second Table.
First proportional after radius,	9999900.
The last proportional.	9995001.224804023027881398897012
Third Table.
Column I.
First proportional after radius.	9995000.
The last proportional.	9900473.578023286050198667424460
Column 2.
The first proportional.	9900000.
Column 69.
The first proportional.	5048858.8878706995 19058238006143
The last proportional.	4998609.401 853 1 893250322338 1 1 730
Half Radius, ....	5000000.
One-tenth of Radius, .	1 000000.

mistake in the Second table, this was not the case with those in the Magnus Canon computed by Ursinus and published in 1624. The logarithm of 30° or half radius, for instance, is there given as 693 147 18 (see specimen page of his Table, given in the Catalogue), which is correct to the number of places given. But in a table of the loga- rithms of ratios (corresponding to the table in sect. 53 of the Constructio), which is given by Ursinus on page 223 of the ' Trigonometria,' the value is stated as 69314718.28, which exceeds the true value by .22. This example will explain how some of the logarithms at the end of the Magnus Canon are too great by i in the units place. Notwithstanding

this,

Notes.

95

Logarithms computed by Napier's Method.	Logarithms computed by Present Method.
1 .000000050000005000000500 100.000005000000500000050000	1.000000050000003 1 00.000005000000333
100.000500003333525000225002 5000.02 5000 I 666762 500 I 1250094	100.000500003333358 5000.0250001 6666791 7
5001.250416822987527739839231 100025.008336459750554796784618	5001.250416822979193 100025.008336459583854
100503.358535014579332632226320	100503.358535014411835
6834228.380380991394618991389791 6934253-3887I745II45I73788I74409	6834228.380380980004813 6934253-388717439588668
693 147 1.805 59946464604 1962236367	6931471.805599453094225
23025850.929940495214660989152136	23025850.929940456840180

this, the Magnus Canon may safely be used to correct the figures in the text and in the Canon of 16 14, as the latter is to one place less.

I find no reference by Ursinus to the discrepancies between the logarithms of the two Canons. The mistake in the Second table may possibly not have been observed by him, as the preparatory tables for the Canons were different.

The mistake was observed by Mr Edward Sang in 1865, when recom- puting in full the preparatory tables of Napier's Canon to 15 places.

It had been previously pointed out by M. Biot, in his articles on Napier in the 'Journal de Savants 'for 1835, p. 255. The following

M 4 translation

96 Notes.

translation of the passage is given in the ' Edinburgh New Philosophical

Journal' for April 1836, p. 285 : —

It has been said, and Delambre repeats the remark, that the last figures of his [Napier's] numbers are inaccurate : this is a truth, but it would have been a truth of more value to have ascertained whether the inaccuracy resulted from the method, or from some error of calculation in its applications. This I have done, and thereby have detected that there is in fact a sligh't error of this kind, a very slight error, in the last term of the second progression which he forms preparatory to the calculation of his table. Now all the subsequent steps are deduced from that, which infuses those slight errors that have been remarked. I corrected the error ; and then, tising his method, but abridg- ing the operations by our more rapid processes of development, calculated the logarithm of 5000000, which is the last in Napier's table, and conse- quently that upon which all the errors accumulate; I found for its value 693 147 1. 808942, whereas by the modern series, it ought to be 6931471. 805599 ; thus the difference commences with the tenth figure.

It has been shown in the foregoing pages that the difference referred to does not really commence until the fifteenth figure.

Numerical errata in the text. — In consequence of what is mentioned above, the figures in the text are in many places more or less inaccurate, but after careful consideration it is thought that the course least open to objection is to give them as in the original.

Different

Notes. 97

Different Methods described in the Appendix for Con- structing a Table of Logarithms in which Log. i==o and Log. 10=1.

The first method of construction, described on pages 48-50, involves the extraction of fifth roots, from which we may infer that Napier was acquainted with a process by which this could be done. The inference is confirmed by an examination of his * Ars Logistica,' at p. 49 of which (Lib. II., * Logistica Arithmetica,' cap. vii.) he indicates a method by which roots of all degrees may be computed. This method of extrac- tion is referred to by Mark Napier in the * Memoirs,' p. 479 seq.^ and a translation is there given of the greater part of the chapter above referred to. A method based on the same principles is given by Mr Sang in the chapter " On roots and fractional powers " in his * Higher Arithmetic,' and these principles are also made use of by Mr Sang in his tract on the * Solution of Algebraic Equations of all Orders/ pub- lished in 1829.

No general method of extracting roots was known at the time, and it does not appear that Napier had communicated his method to Briggs. At any rate, Briggs did not employ the first method described in com- puting the logarithms for his canon.

II.

The second method, described on page 51, is a method suitable for finding the logarithms of prime numbers when the logarithms of any two other numbers as i and 10 are given. This is done by inserting geometrical means between the numbers, and arithmetical means be- tween their logarithms. The example given is to find the logarithm of 5, but as the example terminates abruptly after the second operation, I append the following table from the article on Logarithms in the ' Edin- burgh Encyclopaedia' (1830), which will sufficiently exhibit the method of working out the example, though it is not carried to the same number of places as that in the text.

N The Table.

98

Notes.

THE TABLE.

Numbers.

Logarithms.

A I.OOOOOO B lO.OOOOOO

c = V(ab) = 3.162277

- D = V(bc) = 5-623413 E = V(CD) = 4.216964 F = V(i^E) = 4.869674 G = J(bf) = 5-232991 H= V(fg) = 5.048065

I = V(fh) = 4-958069

K= ^(hi) = 5.002865 L = /s/(ik) = 4.980416 M= V(kl) = 4.991627 N= V(km)= 4.997240

0 = ^(kn) = 5.000052 P = ^(no) = 4-998647

Q = >/(0P) = 4-999350 R= V(oQ)= 4.999701

s = V(or) = 4.999876

T = V(OS) = 4.999963

V = V(ot) = 5.000008 w= V(tv) = 4.999984 X = V(vw) = 4.999997

Y = V(VX) = 5.000003

z = >/(xy) = 5.000000

a 0.0000000 b I.ooooooo c =J(a + b) =0.5000000 d =|(b + c) =0.7500000 e =^(c + d) =0.6250000 f =J(d + e) =0.6875000 g =J(d + f) =0.7187500 h =Kf + g) =0.7031250 i =i(f+h) =0.6953125 k =J(h + i) =0.6992187 1 =J(i + k) =0.6972656 m = J(k + l) =0.6982421 n =i(k + m) = 0.6987304 0 =J(k + n) =0.6989745 p =J(n + o) =0.6988525 q =l(o + P) =0.6989135 r =|(o + q) =0.6989440 s =i(o + r) =0.6989592 t =i(o + s) =0.6989668 V =^(o + t) =0.6989707 w = i(t + v) =0.6989687 X =J(v + w) =0.6989697 y =J(v4-x) =0.6989702 z =J(x + y) =0.6989700

III.

In the description of the third method, on pages 53-54, it is explained that when log. 1 = 0 and log. 10 is assumed equal to unit with a number of cyphers annexed, a close approximation to the logarithm of any given number may be obtained by finding the number of places in the result produced by raising the given number to a power equal to the assumed logarithm of 10. As an example, Napier mentions that, assuming log.

10

Notes.

99

10= looooooooo, the number of places, less one, in the result produced by raising 2 to the iooo.oooDooth power will be 301029995. So that re- ducing these in the ratio of 1 000000000, we have log. 10= i and log. 2 = .301029995 &c. The process is explained by Briggs, pages 61-63, and the first steps in the approximation are shown in a tabular form. The table, extended to embrace Napier's approximation, is given below : in this form it will be found in Hutton's Introduction to his Mathe- matical Tables, with further remarks on the subject.

The method, it will be seen, is really one for finding the limits of the logarithm. These limits are carried one place further for each cypher added to the assumed logarithm of 10, but their difference always remains unity in the last place. Bringing together the successive approximations obtained in the table, we find —

When 2 is raised to the power	The greater limit of its logarithm is
I	I.
10	.4
100	.31
1000	.302
1 0000	.3011
I 00000	.30103
I 000000	.301030
lOOOOOOO	.3010300
I 00000000	.30103000
looooooooo	.301029996

And the less limit is

•3

•30

.301

.3010

.30102

.301029

.3010299

.30102999

.301029995

THE TABLE.

Powers of 2.

Indices of powers of 2.

Number of places in powers of 2.

2

4 16

256

I 2

4 8

I -rl =log. 2

1 » 4

2 „ 16

3 » 256

1024

10486 etc. 10995 »

12089 n

10 20 40 80

4 -7- 10 =log. 2

7 M 4 13 » 16 25 „ 256

N

The Table — confd.

lOO

Notes.

THE TABLE— continued.

Powers of 2.	Indices of powers of 2.	Number of places in powers of 2.
12676 &C. 16069 » & :	100 200 400 800	31 -Moo =log. 2 61 „ 4 121 „ 16 241 „ 256
1071s „ 11481 „ 13182 „ ^7377 »	1000 2000 4000 8000	302-4-1000 = log. 2 603 „ 4 1205 „ 16 2409 „ 256
19950 „ 39803 „ 15843 „ 25099 „	lOOOO 20000 40000 80000	3011-i-IOOOO log. 2 6021 „ 4 12042 J?p 24083 ^
99900 „ 99801 „ 99601 „ 99204 „	lOOOOO 200000 400000 800000	30103 60206 I204I2 240824
99006 „ 98023 „ 96085 „ 92323 »	lOOOOOO 2000000 4000000 8000000	301030 602060 I 204 I 20 2408240
90498 „ 81899 » 67075 » 44990 „	lOOOOOOO 20000000 40000000 80000000	3010300 6020600 I 204 I 200 24082400
36846 „ 13577 „ 18433 » 33977 ,,	lOOOOOOOO 200000000 400000000 800000000	30103000 60206000 I204II999 , 240823997
46129 „	lOOOOOOOOO	301029996

A CATALOGUE

OF THE WORKS OF

JOHN NAPIER

of Merchiston

To which are added a Note of some Early Logarithmic Tables and other Works of Interest

Compiled by

William Rae Macdonald

N 3

PRELIMINARY.

Contents and Arrangement.

The works of John Napier of Merchiston were published in the following order : —

A Plaine Discovery of the Whole Revelation of St John, published in English in 1593.

Mirifici Logarithmorum Canonis Descriptio, published in Latin in 1614, together with the Canon or Table of Logarithms.

Rabdologise, published in Latin in 161 7, the year of the Author's death.

Mirifici Logarithmorum Canonis Constructio, published in Latin in 16 19, two years after the Author's death, by his son, Robert Napier.

Ars Logistica, 'The Baron of Merchiston his booke of Arithmeticke and Algebra,' in Latin, edited by Mark Napier, and published in 1839.

These works naturally fall into three groups : the first contains the result of his early studies in Revelation by which he became famous among the Reformed Churches of Europe, as one of the most learned Theologians of the day ; another contains his works on Logarithms, by which his fame as a Mathematician was established in the scientific world ; between these two groups may be placed his other works, which were more or less preparatory to or suggested during the elaboration of his Logarithms. Accordingly, in the Catalogue we have arranged his works in the following order : I. A Plaine Discovery ; IL Ars Logistica ; III. Rabdologiae ; IV. The Descriptio and Constructio. As a supple-

N 4 ment

I04

Preliminary.

ment are added particulars of the Logarithmic tables computed by Ursinus, Kepler, and Briggs, with a note of some other works of interest.

Collation.

The arrangement of the title-page in the original is indicated by placing an upright bar to mark the end of each line.

The symbols 4°, 8°, 12°, etc., indicate the number of leaves into which the sheet of paper was folded ; but the number of leaves made up in the signatures sometimes differs from this : thus, for example, in the early editions of A Plaine Discovery, though the sheet is folded into 4 there are 8 leaves to each signature.

The measurement of the largest copy examined has been given, but in many cases the work in its original state must have been considerably larger, the copy having been cut down in rebinding.

The signatures in the editions described consist of the letters of the alphabet excluding J, U, and W, or 23 letters (in one or two instances J and U are used for I and V). To each letter belongs a bundle of leaves, 4, 8, 12, &c., as the case may be. The leaves in each bundle are usually numbered thus : — C, C2, C3, etc., but frequently the signatures are printed only on the first one or two leaves in each bundle. The signature is very rarely printed on a title-page. When a leaf is described as B3, for instance, both sides are included, B$^ being used to signify the recto and B32 the verso.

Libraries.

To each entry in the Catalogue, under the head of Libraries, is appended a note of the principal public libraries in this country which possess copies, to these the names of a few foreign libraries are added. The following abbreviations are employed : —

Un. Ab. University,

Un. Camb. University,

Trin, Col. Camb. Trinity College, St John's Col. Camb. St John's College,

Trin. Col. Dub. Trinity College,

Adv. Ed. Advocates,

Sig,Ed, Signet,

Aberdeen. Cambridge.

do.

do. Dublin. Edinburgh.

do.

Un. Ed.

Preliminary.

105

Un. Ed. New Col Ed, Act Ed. Un, Gl,

Hunt, Mus, Gl.

Brit. Mus, Lon, Un. Col. Lon, Guildhall Lon, Roy. Soc. Lon, Lambeth Pal. Lon. Sion Col. Lon. Act. Lon, Chethaiiis Manch, Bodl, Ox/, Qu, Col, Oxf, Un. St And.

Kon. Berlin, Stadt. Bern, Stadt. Breslau,

Un, Breslau,

Kon. Off. Dresden Stadt. Frankfurt, Pub, Genlve, Kon. Hague, Un. Halle, Stadt. Hannover, Un. Leiden,

Maat. Ned. Let, Leiden,

Un, Leipzig,

K. Hof u, Staats

MUnchen, As tor New York, Nat, Paris,

Soc, Prot, Fr. Paris,

Min. Schaffhausen, Un, Utrecht, Stadt. Zurich.

University,

New College,

Faculty of Actuaries,

University,

( Hunterian Museum in the Univer- ( sity buildings,

British Museum,

University College, .

Corporation or Guildhall,

Royal Society, .

Lambeth Palace,

Sion College, .

Institute of Actuaries,

Chetham's Library, .

Bodleian, ....

Queen's College,

University,

Konigliche Bibliothek, .

Stadtbibliothek,

Stadtbibliothek, K Konigliche und Universitats Bibli- ( othek,

Konigliche Offentliche Bibliothek, .

Stadtbibliothek, . . . .

Biblioth^que Publique, .

Koninklijke Bibliotheek, ,

Konigl. Universitats-Bibliothek,

Stadtbibliothek, . . . .

Bibliotheek der Rijks-Universiteit,

!De Maatschappij der Nederlandsche Letterkunde. Library in the Uni- versity buildings, Universitats-Bibliothek, .

K. Hof- und Staats-Bibliothek,

Edinburgh.

do.

do. Glasgow.

do.

London.

do.

do. Manchester. Oxford.;

do. St Andrews.

Berlin.

Bern.

Breslau.

^ do.

Dresden.

Frankfurt a/M.

Geneve.

s'Gravenhage.

Halle a/S.

Hannover.

Leiden.

- do.

Leipzig. MUnchen.

Astor Library, New York.

Biblioth^que Nationale, . . . Paris. rSocidtd de I'Histoire du Protestan- ) , \ tisme Frangais, )

Ministerial Bibliothek, . . . Schaffhausen.

Bibliotheek der Universiteit, . . Utrecht.

Stadtbibliothek, .... Zurich.

O Bibliographies.

:o6 Preliminary.

Bibliographies.

As several works of this kind are mentioned in the Catalogue, a short note of the particular work and edition referred to is given below : —

Messkatalog. — Catalogus universalis pro nundinis Francofurtensibus autumnalibus, de anno mdcxi. Hoc est : Designatio omnium librorum, qui hisce nundinis autumnalibus vel noui vel emenda- tiores et auctiores prodierunt. Das ist : Verzeichnuss aller Bucher, so zu Franckfurt in der Herbstmess, Anno 1611 entweder gantz new oder sonsten verbessert, oder auffs new widerumb auffgelegt, in der Buchgassen verkaufft worden.

Francofurti, Permissu Superiorum, Typis Sigismundi Latomi.

The Frankfurt catalogues were issued for the half-yearly book fairs held in that city at Fastenmesse and Herbstmesse.

In these catalogues, and in bibliographical works founded on them, as those of Draudius, Lipenius, etc., the place and name given cannot be taken as the actual place of publication and name of publisher without corroborative evidence. Thus, for example, the editions of the 'Descriptio' 16 14, * Rabdologiae ' 1617, and the * Constructio ' 161 9, which were published at Edinburgh by Andrew Hart, are sometimes given with the correct particulars, and again appear as issued at Amster- dam, the first by lansonius, and the two others by Hondius. There is little doubt, however, that these were simply importers of the Edin- burgh editions who supplied the German market. Similar remarks apply to the translations and other editions of Napier's works.

Draudius. — Bibliotheca Librorum Germanicorum Classica. Durch M. Georgium Draudium.

Franckfurt am Mayn, Balthasaris Ostern. 1625.

M Bibliotheca Classica sive Catalogus Officinalis. M. Georgi6 Draudio. Fracofurti ad Moenum. Balthasaris Ostern. 1625.

II Bibliotheca Exotica sive Catalogus Officinalis Librorum Peregrinis Linguis usualibus scriptorum, videlicet Gallica . . . Anglica . . . &c., omnium, quotquot in Officinis Bibliopolarum indagari potu- erunt, & in Nundinis Francofurtensibus prostant, ac senales hab- entur. A Frankfourt, Par Pierre Kopf.' 1610.

Another edition, 1625.

Le

Preliminary. 107

Le Z^«^.— Bibliotheca Sacra, Jacobi Le Long.

Parisiis, Apud F. Montalant. 1723.

Freytag. — Analecta Literaria de Libris Rarioribus. Edita a Frider. Gotthilf Freytag, I.C. Lipsiae, In Officina Weidemanniana. 1750.

Gerdes. — Florilegium Historico-criticum Librorum Rariorum. (By Daniel

Gerdes.) ST'"^^I ^P"d {?1„^rP^'' *1 ■763-

& Bremae ) ^ IG. Wilh. Rump. J ^

Rotermund. — Fortsetzung und Erganzungen zu Christian Gottlieb Jochers allgemeinem Gelehrten - Lexiko, worin die Schriftsteller aller Stande nach ihren vornehmsten Lebensumstanden und Schriften beschrieben werden. Angefangen von Johann Christoph Adelung, und vom Buchstaben K fortgesetzt von Heinrich Wilhelm Roter- mund, Pastor an der Domkirche zu Bremen. Fiinster Band.

Bremen, bei Johann Georg Heyse. 1816.

Kayser. — Bucher Lexicon (1750-1832) von Christian Gottlob Kayser.

Leipzig. Ludwig Schumann. 1835.

Ebert. — A General Biographical Dictionary. Frederic Adolphus Ebert.

Oxford. University Press. 1837.

Lowndes. — The Bibliographer's Manual of English Literature, by William Thomas Lowndes. London. Henry G. Bohn. 1861.

Brunei, — Manuel du Libraire. Par Jacques Charles Brunet.

Paris. Firmin Didot, &c. 1863.

Graesse.—Tx€sox de Livres Rares et Prdcieux. Par Jean George Thdodore. Graesse. Dresde. Rudolf Kuntze. 1863.

Z^/;«^C^/.— Catalogue of the Library of the late David Laing, Esq., LL.D., Librarian of the Signet Library (sold in four portions in Dec. 1879, in Apr. 1880, in Jul. 1880, and in Feb. 1881).

Memoirs,— yitmoxxs of John Napier of Merchiston. By Mark Napier.

Edinburgh. Wm. Blackwood. 1834.

o 2

A CATALOGUE

OF THE WORKS OF

JOHN NAPIER

of Merchiston.

I. — A Plaine Discovery of the whole Revelation

of St John.

I. Editions in English.

Edinbvrgh | Printed By Ro- 1 bert Walde-graue, prin- 1 ter to the Kings Ma-|jestie. 1 593. | Cum Priuilegio Regali.|

[On either side of the Title are well executed woodcuts of " Pax" and " Amor."]

4°. Size 7|x6^ inches. Ai is blank except for a capital letter 'A'. A2\ Title. A22, Arms of Scotland and Denmark impaled, for James VI. and his Queen Anne of Denmark; at foot, **/« vaine are all earthlie conivnctions^ vnles Tjve be heires together^ and of one bodie^ and fellow partakers of the promises of God in

O 3 Christ,

I lo Catalogue.

Christy by the Evangell.'' As^-As^ 5 pages, " To The Right Excellent ^ High And Mightie Prince^ lames The Sixty King Of Scottes, Grace And Pecue, dfc. ", ' The Y.^\s\e 'Dedicoloviey^ svgntdi^^ At Marchistoun the 2(^ daye of lamiary 1593. . . . lohn Napeiry Fear of Marchistoun.^' As^-A;^, 5 pages, " To the Godly and Christian Reader y A8\ " The booke this bill sends to the Beast y \ Craning a7nendment now in heasty I" with 26 lines following, then ^'■Faults escaped "y 16 lines. AS^, ^^A Table of the Conclusions introductiue to the Reuelationy and proued in the first TrecUisey Bl^-F3^, pp. 1-69, ** The First And Introdvctory Treatisey conteining a searching of the true meaning of the Reuelationy beginning the discouerie thereof at the places most easiey and most euidentlie knaiuney and so proceeding from the knozvne, to the proouing of the vnknowney vntill finallicy the whole groundes thereof bee brought to lighty after the manner of Propositions. ''\ 36 Propositions and Conclusion. F3^ p. 70, "^ Table Definitive And Diuisiue of the xvhole Revelation.''^ F4^-S7^ pp. 71-269, " The Second And Principal Treatis, wherein {by the former grounds) the whole Apocalyps or Reuela- tion of S. lohn, is paraphrasticallie expounded, historicallie applied, and temporallie datedy with notes on euery difficultie, and arguments on each Chapter "; at the begin- ning of each chapter is " The Argument.'" , then follow " The Text."", ^* Paraphrastical exposition.", "Anno Christi."y and *' J/istorical application.", the four subjects being arranged in parallel columns (in chapters i to 5, and 7, 10, 15, 18, 19, 21, and 22, there is no Historical application, in which case the columns for it and also for Anno Christi are omitted), at the end of each chapter ** Notes, Reasons, and Amplifications." are added. S72-S82, 3 pages, " To the misliking Reader whosoeuer.'' T1I-T42, 8 pages, "Hereafter ^olloweth Certaine Notable Prophecies agreable to our purpose, extract out of the books of Sibylla, whose authorities neither being so authentik, that hitherto we could cite any of them in matters of scriptures, neither so prophane that altogether we could omit them : We haue therefore thought very meet, seuerally and apart to insert the same here, after the end of this worke of holy scripture, because of the famous antiquitie, approued veritie, and harmonicall consentment thereof with the scriptures of God, and specially with the i8. chapter of this holy Revelation."

Signatures. A to S in eights + T in four = 148 leaves.

Paging. 16 + 269 numbered +11 = 296 pages.

Errors in Paging. Page 26 numbered 62, and page 229 numbered 239.

The outside sheet (leaves i, 2, 7, 8) of Signature B was set up a second time, with sHght differences in the spelling and occasionally in the division into lines. Consequently copies may be found in which the title of the First treatise does not agree exactly with that given above. The Advocates' Library in Edinburgh has copies of the two varieties.

The following extract explains the circumstances under which this first work of Napier's was published. The passage begins at the second last line in the second page of the address ' To the Godly and Christian Reader.' (In the edition of 161 1 the passage begins on line 7 of the third page.)

After

Catalogue. i i i

After the which, although (greatly rejoycing in the Lord) I began to write thereof in Latine : yet, I purposed not to haue set out the same suddenly, and far lesse to haue written the same also in English, til that of late, this new insolencie of Papists arising about the 1588 year of God, and dayly increasing within this //and doth so pitie our hearts, seeing them put more trust in lesuites and seminarie Priests, then in the true scripturs of God, and in the Pope and King of Spaine, then in the King of Kings : that, to preuent the same, I was constrained of compassion, leaning the Latine, to haste out in English this present worke, almost vnripe, that hereby, the simple of this Ilaiid may be in- structed, the godly confirmed, and the proud and foolish expectations of the wicked beaten downe, [purposing hereafter (Godwilling) to publish shortly, the other latin editio hereof, to the publike vtilitie of the whol Church.] What- soeuer therfore through hast, is here rudely and in base language set downe, I doubt not to be pardoned thereof by all good men.

The passage enclosed in square brackets is omitted in the edition of 161 1 (also in that of 1645) and in its place is inserted the following passage.

And where as after the first edition of this booke in our English or Scottish tongue, I thought to haue published shortlie the same in Latine (as yet God- willing I minde to doe) to the publike vtilitie of the whole Church. But vnder- standing on the one part, that this work is now imprinted, & set out diuerse times in the French & Dutch tongs, (beside these our English editions) & therby made publik to manie. As on the other part being aduertised that our papistical, adversaries wer to write larglie against the said editions that are alreadie set out. Herefore I haue as yet deferred the Latine edition, till hauing first scene the aduersaries obiections, I may insert in the Latin edition an apologie of that which is rightly done, and an amends of whatsoeuer is amisse.

We see from the above that in 161 1 Napier still had the intention of publishing a Latin edition, but this idea had, no doubt, to be given up owing to the demands made on him by his invention of Logarithms.

Libraries, Adv. Ed. (both varieties); Sig. Ed.; Un. Gl. ; Mitchell Gl.; Un. Ab.; Un. St And.; Brit. Mus. Lon.; Bodl. Oxf.; Qu. Col. Oxf.; Un. Camb. ; Trin. Col. Camb. ;

O 4 Cer-|taine

112 Catalogue.

Cer-|taine Oracles of Sibylla, agreeing | with the Revelation and other I places of Scripture. | Newlie Imprinted and corrected. |

Printed For lohn Norton Dwel-|ling in Paules Church-yarde, neere vnto | Paules Schoole. | 1 594. |

4°. Size 7f X 5| inches.

This edition is very like that of 1593, only the ornamental Title-page has been superseded by a plainer one, the ornament appearing in 1593 at the head of the Epistle dedicatorie now doing duty at the head of the Title-page. The collation remains the same, except as regards the spelling, and also that on Signature A8^ the * Faults escaped ' are now omitted, being corrected in the text. The type is the same, but has been reset, there being numerous differences in spelling and occasional slight differences in the division into lines. The headpieces employed are, with one exception, found in the edition of 1593, but they are less varied and are frequently used in different places. It seems highly probable that this edition was printed in Edinburgh by Waldegrave for John Norton.

Libraries, New Col. Ed.; Brit. Mus. Lon.; Bodl. Oxf.;

Edinbvrgh, I Printed by Andrew Hart. i6ii.|Cum Privilegio Regiae Maiestatis.|

4°. Size 6| X h\ inches. Ai^, TUle. A12-A4I, 6 pages, ' To the Godly ....

Reader; and ♦ The book this bill ' A42, Table B1I-H22, pp. 1-92, The first

Treatise. H3I, Table. Hs^-YS^ pp. 94-327, The second Treatise. YS^, blank. Zi^-Zz^, pp. 329.332, * To the mislykiiig Reader . . . .' Z3i-Bb32, pp. 333-366,

** A Resolvtion, of certaine doubts, proponed by well-affected brethren, and needfull to

be

Catalogue. i 1 3

be explained in this Treatise" seyen Resolutions. Bb4*-Bb8\ pp. 367-375, Oracles of Sibylla, Bb82, blank.

Signatures. A & B in fours + C to Z and Aa to Bb in eights = 192 leaves.

Paging. 8 + 375 numbered + l = 384 pages.

Errors in Paging. Page 56 numbered 65, and page 299 not numbered.

In this edition the Arms, &c., on back of the title-page, and the Dedication to King James, are omitted, and for the first time the * Resolution of Doubts' appears.

Libraries. Adv. Ed. ; Sig. Ed.; Un. Camb.;

London, I Printed for lohn Norton. 1611. |Cum Privilegio Regiae Maiestatis. 4°. Size 1^ X 5| inches.

This edition is in every respect identical with the preceding, except that the last paragraph of the title-page has been reset, the four words ^^ Edinbvrgh. . . , by Andrew Hart" being altered to ^^ London. . . . for Lohn Norton." The printing of both editions appears to have been done in Edinburgh by Andrew Hart ; his type, head-pieces, &c., being employed in both. The two slight errors in pagination remain as before.

Libraries. Adv. Ed.; Sig. Ed.; Un. Ab.; Bodl. Oxf.; Astor New York;

P phrastically

114 Catalogue.

phrastically and Historically! to the Text.| By John Napier, Lord of Marchiston. | With a Resolution of certain doubts, | moved by some well affected brethren. |Whereunto are annexed certain Oracles of | Sibylla, agreeing with the Revelation,] and other places of Scripture. I And also an Epistle which was omitted in| the last Edition. | The fifth Edition : corrected and amended. |

Edinbvrgh, | Printed for Andro Wilson, and are to be sold at his I shop, at the foot of the Ladies steps. 1645.]

4°. Size 7i X 6| inches. Leaf i^, Title, i^ blank. 2^-3^, 3 pages, Dedi-

cation to King James. 3^-5^ 5 pages, To the Godly .... Reader. 6^, * The Book this Bill . . . .' 62, Table. B1I-I3I, pp. 1-61, The first Treatise. I32, blank. I4I, p. 63, Table. I42, blank. Ki^-Iia^, pp. 65-244, The second Treatise. Aaai^- Aaa2^, pp. 1-3, To the misliking Reader. Aaa22-Ddd42, pp. 4-32, A Resolution of Doubts. Eeei^-Eee42, pp. 31-38 [33-40], Prophecies of Sibylla. (In some copies an additional sheet is inserted with list of Errata, see Note.)

Signatures. [A] in six (leaves 4 & 5 are an insertion) + B to Z and Aa to Hh in fours + Ii in two + Aaa to Eee in four= 148 leaves.

Paging. 12 + 244 numbered + (38 + 2 for error = ) 40 numbered = 296 pages.

Errors in Paging. In pp. 1-244 there are 10 errors which do not affect the total ; but in pp. I -[40] the numbers 31 & 32 are twice repeated, so that numbers on all the subsequent pages are understated by 2.

In Glasgow University Library is a copy of this edition with an extra leaf inserted at the end containing ** Errata. — Curteous Reader thou art desired to correct these faults following^ which chiefly happened through the absence of the Author and the difficulty of the Coppy. viz." this is followed by ten lines of corrections.

The author's name, it will be observed, is spelt on the title-page in the modern form, and the Dedication to King James is signed, ^^/ohn Napier^ Peer of Marchiston." The substitution of Peer for Fear or Feuer of Merchiston seems to have been intentional. It is not noticed in the errata, but is of course a mistake.

This is the only edition in which " The Text.", " The Paraphrasticall Expositions^ and the ^^ Historicall Application." , are printed successively and not in parallel columns. The " Historicall Application." , is printed in black letter. " An. Chr." is printed on the margin of each page in both treatises.

Libraries. Adv. Ed.; Sig. Ed.; Un. Ed.; New Col. Ed. (2); Un. Gl. (2); Un. Ab.; Brit. Mus. Lon.; Sion Col. Lon.; Un. Camb.; Trin. Col. Camb.;

2. Editions

Catalogue. 115

2. Editions in Dutch.

Middelburgh | By Symon Moulert, woonende op den | Dam inde Druckerije. Anno i6oo.|

4°. Size 7|x5^ inches. Black letter with exception of the pages from

* i^ to * 3^ and a few passages here and there. * i^, Title-page. * i^,

" Extract wt de Privilegie " granted to M. Panneel for lo years by " Z>^ Staten Generael der vereenichde Nederlanden" signed at *^ s' Graven- Haghe, den 4. Augustt. 1600. . . . ." At the foot of the page are three lines of errata under the heading, *' Som- mighe fatiten om te veranderen.''"' * 2^-* 3^, 4 pages, ^^Aende E. E, Wyse Endt

Voorsienige Heeren^ Myne Heeren, Bailliv Burghemeesteren, Schepenen, ende Raedt der vermaerder Coopsiadt Middelburgh in Zeelandt,^^ signed " Tot Middelburgh in Zeelandt, desen 20. yiclij, inden Jare Christie 1600. V. E, E. Onderdanighe dienst-willige, M. Panneel. ^^ * 4^-* * i^ 4 pages, " Den Seer Wtnemenden hooghen ende Machtighen Prince Jacobo de seste Coninck der Schotten ghenade ende vrede, &'c.f" signed ** Tot Marchistoun Den 29. dagh Januarii 1593, uwe Hoocheyts seer ootmoedighe ende ghehoorsaem ondersaet JOHAN Napeir. Erfachtich Heer van Mar- chistoun.''^ **2^-**4^, 5 pages, ^^ Aen den Godtsalighen ende Christelijcken Leser." * *4^ *' T'boeck sent dit schrift de beeste^ of zijt woude noteren^ \Begee- rende dats t^meeste, datse haestelijck vvil bekeeren.y\ followed by 26 lines. On a folding sheet preceding Ai^ is ^^ Een tafel vande inleydende sluytreden^ deser open- haringe bewesen int e erste tractaet." Ki^-^z^ pp. 1-68 (last 4 pages not, num- bered), ^^ De erste ende het inleydende Tract aet ofte handelinge Inhoudende een onder- soeck van den rechten sin ofte meefiinghe der Openbaringhe Joannis d'openinghe van dien beginnende aende plaetsen die lichst om verstaen ende best bekent zijn ende also voortgaende vande bekende tot D'onbekende tot dat den gantschen grondt daer van eyn- delinghe int licht ghebrocht iverdt ende dat by maniere van PropositienJ" This Treatise contains the 36 Propositions, and on Aai^ is the '^ Beslvyt'' or Conclusion. Aai^ [p. i], '■* Een verclarende en afdeelende Tafel vande gheheele openbaringhe.^'' Aa2i-Ggg3^, pp. 2-237, ^^ Het tweede ende voomaeviste Tractaet daer in (achtervol- ghende de voorgaende grontreden) t'geheele Apocalipsis ofte openbaringe des Apostels Joannis op paraphrastischer wijse wtgheleyt op historischer ivijse toegheeygent en

p 2 tijdelijck

ii6 Catalogue.

tijdelijck gedcUeert wort. Met aenwijsinghen op elcke swaricheyt ofte hinderinghe ende argument op elck CapitteV\ the chapters commence with ^^ Het Argument^^"* then follow in four parallel columns *^ Den Text", *^ Paraphrasis", **Anno Christi"^ and ^* Historie^^ (the 3d and 4th columns are wanting in the chapters mentioned in the Edin. 1593 edition), at the end of the chapter are added ^^Aenwijsinghen Redenen ende breeder VerclaringhenJ^ Ggg'i^-H.\i\i2^, 6 pages, " Tafel ofte Register der

aenwistinghen Redenen ende breeder verclaringhen" an alphabetical index of the principal matters contained in the work. Hhh2^-Hhh32, 3 pages, '* Totten

Leser, " which appears to be a Glossary of certain words used in the work. Hhh4^

** Errata inde Propositien" followed by 15 lines of corrections. Hhh4^, blank.

Si^atures. * and * * and A to H in fours + J in two + Aa to Zz and Aaa to Hhh in fours = 166 leaves.

Paging. 16 + 68 numbered (except last 4) + 237 numbered (except first 3) + 11 = 332 pages.

Errors in Paging. There are some 18 of these, mostly in the second part, but none of importance.

This translation by M. Panneel omits the address To the Mislyking Reader^ and the Oracles of Sibylla^ but otherwise it appears to be a full translation of the edition of 1593.

Graesse states that there is an edition, ** trad, en hollandais par M. Pannel: Avist. 1600 in 8°." Most likely this is the edition referred to.

Libraries. Guildhall Lon. ; Stadt. Zurich;

Een duydelijcke verclaringhe | Vande gantse Open-|baringe loannis des Apostels.|T'samen ghestelt in twee Trac-|taten: Het eene ondersoeckt ende bewijst de wa-|re verclaringe der selver. Ende het ander appliceert ofte|voecht, ende eyghentse Paraphrastischer ende|Historischer wijse totten Text. | Wt-ghe- gheven by lohan Napeir, Heere|van Marchistoun, de Ionghe.| Over-gheset vvt d'Enghelsche in onse Nederlandtsche|sprake. Door I M. Panneel, vvijlent Dienaer des H. vvoords Godts|tot Middelburch. | Den tweeden druck oversien, ende in velen plaetsen verbetert. I Noch zijn hier by-ghevoecht vier Harmonien, &c. van nieus over- 1 gheset wt het Fransche. |

Middelburch, I Voor Adriaen vanden Vivre, Boeck-vercooper, | woonende inden vergulden Bybel, Anno 1 607. | Met Privilegie voor 10 Iaren.|

8'. Size

Catalogue. i i 7

8*. Size 6| X 4i inches. Black letter, except from * 2 to * 7. * i^ Title-

page. * 1 2 blank. *2i-*42, 6 pages, ^'Aende E. E. VVyse Ende Voorsienige

Heeren, . . . ." ^S^-^'J^S^zge^,'* J)enseerwt-nemenden,Hooghenettde Machii- ghen Prince lacobo . . . ." *8i-**4i, 9 pages, '' Aen den Godtsalighen ende Christelijcken Leser.'* * * 4^, ^^T^boecks ettdi dit schrift der Beeste, ,en bidt dat syt noteere, \ Op dat sy haer {dit's t'meeste) soo't moghelijck is bekeere, \ " followed by 28 lines. First Table wanting. Ai^-Fs^ pp. 1-89, *'ffet eerste ende inky dende Tractaet oftehandelinghe . . , ."^the36 propositions and the ^* Beslvyt" ox conclusion. Second Table wanting. F52-Aa4''^, pp. 90-376, " Het tweede ende voornaemste Tractaet^

daer in {achter volghende de voorgaeitde gront-reden) fgheheele Apocalipsis ofte Openbar- inghe des Apostels loannis, op Paraphrastischer wijse Twtgeleydt, ende op Historischer wijse ende nae de tijden der gheschiedenissen toe-gheeyghent wordt : Verciert met aen- wijsinghen op elcke duystere plaetse, ende met Argument op elck CapitteV Aa5^-Aa8^, PP« yn-Z^Zi ^^ Aen deti Leser^ wien dit werck mishaeght." AaS^, blank.

BbiS Title-page. Bbi2, blank. Bh2^-'Ehf, 4 pages, ** Voorreden." signed " Gre- orgivs Thomsson. ". Bb4^-Dd2^, 29 pages, contain the Vier harmonien. TiAj^- Dd42, 5 pages, " Tafel vande principaelste materien die int geheele Boec verhandelt werden soo in de Propositien ah in de Aenivijsinghen achter yder Capittel.''* At the foot of the last page {T>d/^^) is printed: " Tot Middelbvrch,\Ghedruckt by Symon Moulert, Boeck-ver cooper \woonende op den Dam, inde Druckerije. Anno 1607. | "

Signatures. * in eight + * * in four + A to Z and Aa to Cc in eights + Dd in four = 224 leaves.

Paging. 24 + 383 numbered + i + 40 = 448 pages.

Errors in Paging. Pages 143, 187, 269, and 308 numbered in error 144, 189, 270, and 208 respectively.

On comparing this edition with that of 1600, we find that the ad- dress To the Mislyking Reader is now given, and there is also added a translation of the Quatre Harmonies, from the French editions of 1603 et seq. Further, we find, besides the usual differences in spelling, occasional alterations in the translation. For example, compare the wording, &c., in signatures * *42 and F52 of the above collation with that corresponding in the signatures * * 4^ and h.2^2^ of the collation of

P 3 the

ii8 Catalogue.

the 1600 edition. From this it would appear that for this 1607 edition the translation of t6oo was revised, possibly by G. Panned, the trans- lator of the Quatre Harmonies. Both the Tables are wanting in the copy examined.

Libraries, Maat. Ned. Let. Leiden :

3. Editions in French.

Va, pren le liuret ouuert en la main de I'Ange. Apoc. 10. 8. | Hola Sion qui demeures auec la fille de Babylon, sauue-toi. Zach. 2. 7. | le te conseille que tu achetes de moy de I'or esprouue par le feu, afin que tu | deuiennes riche, | Et que tu oignes tes yeux de coUyre, afin que tu voyes. Apoc. 3. 18. | Qui lit, I'entende. Matth. 24. 15.

A La Rochelle. I Par lean Brenovzet, demeurant pres|la bou- cherie Neufue.| i6o2.|

4°. Size 9 X 6i inches. ai^. Title, a 1 2, blank. a2i-a3i, 3 pages, "^ Tres-

havt Et Tres-pvissant laqves Sixiesvie, Roy D'escosse. Gr. (Sr" P." signed ^^ lean Noti- pareil,'''' ^"^-^2^^ 6 pages, 'Mz; Lectevr Pievx Et Chrestieti" 62^ & 63^,

** Avx Eglises Francoises Reformees Tant En La France Qv'aillevrs 6".", signed "Georges Thomson." 63^, Poems — ^^ De Georgii Thoinsonii Paraphrasi Gallica

Ad Galliam. Ode" 40 lines; also *^ Idem,'^ 8 lines, signed ^'' loatines Duglassms Musilburgenus.''^ Preceding Ai^ on a folding sheet is " Table des propositions seru- antes d" introduction h V Apocalypse prouuees an premier Traite, lesquelles sont couchees en ceste table selon leiir ordre naturely mais au premier traite suiuant sont mises selon Vordre de demonstration afin que chaque proposition soil prouuee par la pj'ecedante." Ai^-Gi^, pp. 1-50, "Z^ Premier Traite Servant D'introdvction, Contenant Vne recerche du vray sens de V Apocalypse, commenfant la descouuerture d'icelle par les points les plus aises ^jnatiifestes, dr" passant dHceux d, la preuue des incognuSy itisques a ce que finalement tons les points fondamentaux soyent esclaircis par forme de propositions." 36 Propo- sitions

Catalogue. 119

sitions and " Conclvsion." Before Gi^ on a folding sheet is " Table difinissatiie

6- diuisante toute V Apocalypse.'' G2^-Ffi2, pp. 51-234 [226], '' Le Second Et

Principal Traite Avqvel {Selon Les Fondemens Desja posez) toute V Apocalypse est paraphrastiqicement interpretk^ <Sr= appliquee aux tnatieres, selon leur histoire, df datee du temps, auquel chaque chose doit arriuer, atiec anjtotations sur chaque difficulte, <Sr» argumens sur chaque chapitre." ; at the beginning of each chapter is ^^ VArgvment'\ followed by "Z« Texte", ^^ L' Exposition Paraphrastique", *^An de Christ", and '*£' Application historique" in four parallel columns (the 3d and 4th of which are wanting in the chapters mentioned in the Edinr- 1593 edition), and at the end of each chapter are ^'■Annotations, Raisons, ^Amplifications" rf2^-Ff32, pp. 235-238

[227-230], are ^^ Av Lectevr Mai -content." Yi^-Vxi?; 19 pages, " Table De

Tovtes Les Matieres Primipales Contenves, Tant an pre77iier qu'au second Traite sur V Apocalypse" arranged alphabetically; at foot of last page ^^ Fautes suruenues en V impression." 4 lines. lii^, blank.

Signatures, a in four + e in three (leaf €4 being cut out), + A to Z and Aa to Hh in fours + Ii in one =132 leaves.

Paging. 14 + (238-8 for error = ) 230 numbered + 20 = 264 pages.

Errors in Paging. Numbers 8i to 90 omitted, and 98 & 137 twice repeated = - 10 + 2= -8.

The Tables are on two folding sheets which precede Ai^ and 02^,

In all the French editions, the lines " The Book this bill sends to the Beast . . ." and * The Oracles of Sibylla^^ are omitted. The addition here made to the title of the First Table appears in the English edi- tions as a note at the end of the Table.

Libraries. Adv. Ed.: Nat. Paris:

Va, pren le liuret ouuert en la main de I'Ange. Apoc. 10. 8. | Hola Sion, qui demeures auec la fille de Babylon, sauue-toi. Zach. 2. 7. | Je te conseille que tu achetes de moy de Tor esprouue par le feu, [ afin que tu deuiennes riche. | Et que tu oignes tes yeux de colly re, afin que tu voyes. Apoc. 3. 18. | Qui lit, I'entende. Matth. 24. 15.

A La Rochelle,|Pour Timothee lovan.jM. DC. IL|

P 4 This

I20 Catalogue.

This can in no sense be considered another edition, Brenouzet's title- page having simply been cut off half an inch from the back and the above substituted. This substituted title has an ornamental border round the type, whereas Brenouzet's has simply a line. The copy examined for this entry (from Bib. Pub. de Genbve) differs, however, from that examined for the previous entry (from Adv. Lib. Ed.) in certain small points which may be noted, namely: on e$^ the signature is omitted, on e^^ three little ornaments are omitted, on p. 3 the number is omitted, and finally, the principal error in paging commences here with p. 80 being numbered 90 instead of as above, with p. 81 being numbered 91.

Libraries. K. Hof. u. Staats. MUnchen ; Pub. Geneve ; Stadt. Bern ;

II te faut encores prophetizer ^ plusieursf peuples, & gens, & langues, & Rois.| Apoc. 10. ii.[

A La Rochelle,|Par les Heritiers de H. Haultin.|M. DC III.|

8°. Size 7x4f inches, ai^, Title, ai^, blank. a2i-a5^ 7 pages, Dedication to King James. aS^-ej^, 13 pages, Av Ledevr pieux . .*. . €4^-65^, 3 pages, ^^ Avx Eglises Francoises . . . ." es^-eS^, 6 pages. Poems, eight more being added to those in the 1602 edition. eS^, " Aduertissetnent du Translateur au Lecteur.^^ Before Ai^ 7a^/^ on folding sheet. A1I-F6I, pp. 1-91, The first Treatise. F62, blank. Before F;!, Table on folding sheet. F7^-V82, pp. 93-318 [320], The second

Treatise. X1I-X42, 8 pages, Av Lectevr Mal-content. X5^-Y7^, 21 pages,

" Table Des Matieres Primipales Contenves en ce livre." Y72, *^A UEglise. Son- net:' Y8, blank.

Qvatre I Harmonies I Svr La Revelation I De S. lean: Tovchant La I Royavte Prestrise, | & Prophetic de lesus | Christ. | Contenantes

aussi

Catalogue. i 2 1

Zii, Title. Zi2, blank. Zi^^-Zi^, 6 pages, "Za Preface:' Zs^-AaS*, pp. 1-24, The Work itself. At foot of p. 24 is printed, '^ Acheui cT imprinter le premier iour de VAn 1603."

Signatures, a and e and A to Z and Aa in eights = 208 leaves.

Paging. 32 + (318 + 2 for error = ) 320 numbered + 32 + 8 + 24 numbered = 416 pages.

Errors in Paging. The numbers 143 and 144 are twice repeated.

The two Tables are on folding sheets which precede Ai^ and Fy^

Libraries. Un. Ed.; Un. Gl.; Nat. Paris; Kon. Berlin;

II te faut encores prophetizer k plusieurs peuples, | & gens, & langues, & Rois. ) Apoc. 10. 1 1. 1

A La Rochelle,|Par Noel De la croix. 1605.I

8°. Size 7x4^ inches. ai^ Title, ai^, blank. z.2}-z^, 5 pages, pedication to King James. a4*-a82, 9 pages, Av Lectevr pieux .... ei, 2 pages, Avx

Eglises Francoises 62^-64^, 5 pages. Poems y as in the edition of 1603.

€42, Aduertisseitient Before Ai\ Table on folding sheet. Ai^-Fs^,

pp. 1-90, The first Treatise. Before F6S Table on folding sheet. Y&^-Qz^, pp. 91-446 [406], The second Treatise. Cc4^-Cc62, 6 pages, Av Lectvr Mal- content. Cc7^-Ee5\ 29 pages, Table des Matieres Ee52, Sonnet.

Qvatre I Harmonies I Svr La Revelation | De S. lean; Tovchant La I Royavtd Prestrise, | & Prophetic de lesus | Christ. | Contenantes aussi la Prophetic & Histoire Chrcsticnnc|aucunement depuis la naissance de Christ iusques |^ la fin du monde, sans interrup- tion|des visions.|Par G.T.E.||i6oS.|

Q Ee6S

122 Catalogue.

Ee6\ Title, EeS^, blank. Ee7i-82, 4 pages, La Preface. Ffii-GgSi, pp. 1-3 1, The Work itself. At foot of p. 31 is printed, ^^ Acheui (P imprinter le huictiesme iour de luin 1605." GgS^, blank.

Signatures, a in eight + e in four + A to Z and Aa to Gg in eights = 252 leaves.

Paging. 24 + (446-40 for error =) 406 numbered + 36 + 6 + 31 numbered+ 1 = 504 pages.

Errors in Paging. P. 15 is numbered 16, and there are several errors in signature E, but the only error affecting the last page, is p. 401 numbered 441, and so to the end.

The two Tables are on folding sheets which precede Ai^ and F6^.

It will be observed that this is described as * Edition seconde * as well as that of 1603.

Libraries. Adv. Ed.; Un Ed.; Un. Ab. ; Brit. Mus. Lon.; Chetham's Manch. ; Trin. Col. Dub.; Nat. Paris; Stadt. Frankfurt;

II te faut encores prophetizer ^ plusieurs peuples, | & gens, & langues, & Rois. | Apoc. 10. II. I

A La Rochelle, | Par Noel de la Croix. | cId. IoC. VII. |

8**. Size 6fx4i inches. Ai^ Title. Ai^, blank. A2i-A4S 5 pages, Dedication to King James. A^-A'j'^, 6 pages, Av Lectevr Pievx ..... A72-A8^, 2 pages,

Avx Eglises Francoises A8^-B22, 5 pages. Poems, as in the edition of

1603, also the Aduertissement. .... Preceding B3^, Table on folding sheet.

B3^-G72, pp. 1.90, The first Treatise. Preceding GS^, Table on folding sheet. GS^-Dds^, pp. 91-406, The second Treatise. Dde^-DdS^, 6 pages, '' Av Lectevr Mal-content." Eei^-EeS^, 16 pages, Table des Matieres . . . . , on the last page at the end is the Sonnet.

Qvatre I Harmonies I Svr La Revelation De | S. lean : Tovchant La Royav- 1 te, Prestrise et Prophe- 1 tie de lesus Christ. | Con- tenantes aussi la Prophetie & Histoire Chrestienne | aucunement

depuis

Catalogue. 123

depuis la naissance de Christ lus- 1 ques k la fin du monde, sans interru- 1 ption des visions. | Par G.T.E. | do. IdC. VII. |

Ffii, nae. Ffi2, blank. Ff2i-Ff32, 4 pages, La Preface, Ff4i-Hh3S pp. 1-31, The Work itself, Hh32-Hh4S 3 pages, blank.

Sigfiatures, A to Z and Aa to Gg in eights + Hh in four =244 leaves. Paging. 20 + 406 numbered + 22 + 6 + 31 numbered + 3 = 488 pages. Errors in Paging. P. 397 numbered in error 367, and pp. 401-404 numbered in error 441-444. (Quatre Harm.) p. 3 numbered in error 5.

In the title-page of the Quatre Harmonies, the fifth line ends with " Prophe " ; in the Oxford copy this is followed by a hyphen, but in the Breslau and Dresden copies the hyphen is wanting.

Libraries, Bodl. Oxf. ; Stadt. Breslau ; Un. Breslau ; Kon. Off. Dresden ;

II te faut encores prophetizer h. plusieurs peuples,|& gens, & langues, & Rois. | Apoc. 10. ii.j

A La Rochelle,|Par Noel de la Croix. | do. loC. VII. |

8". Size 6| X 4| inches. AiS Tille. Ai\ blank. A2i-A4S 5 pages, Dedication to King James. A^^-A^\ 6 pages, Av Lectevr Piez'x .... A73-A8S 2 pages, Avx Eglises Francoises .... AS^-Ba^, 5 pages, PoemSt as in the edition of 1603, also the Aduertissement .... Preceding B3\ Table on folding sheet. ^3^-052, pp. 1-86, The first Treatise. Preceding G6S Table on folding sheet. G6i-Cc6\ pp. 87-391, The secottd Treatise, four lines are carried over to the top of the page following 391. Cc62-Ddi\ 6 pages, '' Av Lectevr Mal-content:' Ddi^- DdS^, 14 pages. Table des Matieres .... DdS^, Sonnet.

Qvatre I Harmonies I Svr La Revelation De|S. lean : Tovchant La Royav-|te, Prestrise Et Prophe- 1 tie de lesus Christ. |Con- tenantes aussi la Prophetie & Histoire Chrestiennejaucunement

Q 2 depuis

1 24 Catalogue.

deptiis la naissance de Christ ius-|ques k la fin du monde, sans interru-lption des visions. | Par G.T.E.|cIo. loC. VII.|

EeiS Tifle. Eei^, blank. Eea^-EesS, 4 pages, La Preface, ^Q/^-Q^gf, pp. 1-31, The Work itself. ^gf-^g^, 3 pages, blank.

Signatures. A to Z and Aa to Ff in eights + Gg in four =236 leaves. Paging. 20 + 39 1 numbered + 21 + 6 + 31 numbered + 3 = 472 pages. Errors in Paging. These are numerous, especially in signatures L and S, but none affect the last page. The two Tables are on folding sheets which precede B3^ and G6^.

For some reason the type for the Rochelle issue of 1607 was twice set up. In this variety it will be observed that the number of pages occupied by the body of the work is about four per cent less than in the variety described in the preceding entry. The above collation is from the Edinburgh copy. The Paris copy agrees with it, except that the word " Harmonies " in the seventeenth line of the first title-page commences with a small h, as in the previous entry.

An edition * A Genbve chez laques Foillet, 1607, in 8°,' is mentioned by Freytag in the Analecta Literaria, p. 1136. A similar entry, but omitting *Genbve,' is made by Draiidius in the Bibliotheca Exotica, 1625 edition, p. 11. Possibly Foillet was only the introducer of the work at the Frankfurt Book Fair.

Le Long mentions an edition, Geneva, 1642, in 4°. An entry was found, in a library catalogue, under Napier's name, which appeared to substantiate Le Long's statement. In that case, however, the work proved to be the ' Ouverture des secrets de I'Apocalypse de Saint Jean, contenant tres parties .... par Jean Gros. Geneve, Fontaine, 1642, in 4°.'

Libraries. Adv. Ed.; Soc. Prot. Fr. Paris; Stadt. Zurich;

4. Editions

Catalogue. 125

4. Editions in German.

[Note. — In the German editions, the letters printed below as a, o, ii, are in the original printed a, o, u, an earlier way of expressing the 'umlaut.']

Entdeckung aller Geheimniissen in der | Apocalypsi oder Offenbarung S. Jo- 1 hannis begriffen. | Darinen die | Zeiten vnd Jahren der Regierung desz Anti- 1 christs, wie auch desz Jiingsten Tages, so eygentlich | durch gewisse gegriindete Vrsachen auszgerechnet, dasz man fast | nicht dran zweiffeln kan. I Zuvor zwar niemals gesehen noch gehort, wiewol von vie- len vornehmen, gelahrten vnnd erleuchteten Mannern, wie | von dem seligen Mann D. Luthero selbsten, ge- 1 wiindschet worden. | Von I lohanne De Napeier, | Herrn de Merchiston, erstmals in Scotischer Sprache aus | Liecht gegeben. | Jetzt aber treuwlich verdeutschet, | Durch | Leonem De Dromna. |

Dan. 12. I Vnd nun Daniel verbirg diese Wort, (vom Reich vnnd Zeit desz | Anti- christs, vnd desz Jiingsten Tages) vnd versiegele diese Schrifft, bisz aufF die | bestimpte Zeit, so werden viel driiber kommen, vnd grossen Verstand finden. |

Gedruckt zu Gera, durch Martinum Spiessen. | Im Jahr 1611. |

[Printed in red and black.]

4°. Size 1\ X 6i inches. Black letter. (:) i^. Title. (:) i^, blank. (:)

2^-(:J 3^ 4 pages, The Preface *'An den guthertzigen Leser" signed " /w Jahr 1611. Leo de Dromna.^' (:) /^-)( )(2\ 6 pages, ** Register aller vnnd jeder Propositionen^ so in diesem Biichlein tractiert werden '\ being the titles of the 36 propositions con- tained in Napier's First treatise; at the end are added ^^ Errata Typographical^ 18 lines. Ai^-Y2^,^^. i-iT I, The first treatise. ¥2^^, blank.

Signatures. (:) in iom + )( )( in two + A to X in four + Y in two =92 leaves.

Paging. 1 2 + 1 7 1 numbered + 1 = 1 84 pages.

This edition contains a translation by Leo de Dromna of the 36 propositions of Napier's First Treatise, but without its Title and * Con- clusion.' The other parts of the original work are all omitted.

Le Long catalogues editions Leipsic 161 1 and Gera 161 2. Drau- dius also, in Bibliotheca Librorum Germanicorum Classica, p. 290, mentions a Leipsic edition of 161 1. He gives the title exactly as above so far as the words * zweiffeln kan ' at the end of the eighth line,

Q 3 ^ft^*-

126 Catalogue.

after which he adds * ausz Scotischer Sprach verteutscht durch Leonem de Dromna. Leiptzig bey Thoma Schiirern, in 4. 161 1/ These par- ticulars are copied verbatim from the Frankfurter Messkatalog vom Herbst 161 1, sheet Di^. The appearance of Schiirer's name may, how- ever, imply simply that he brought the work to market and issued it at the Frankfurt Book Fair, not that there is an edition bearing on its title-page to have been issued by him at Leipsic.

An edition Gera 1661 in 4° is mentioned by Graesse. His particu- lars regarding German editions of * A plaine discovery * appear to be copied from Rotermund, vol. v. p. 494, where the same date is given ; but there is Httle doubt it is a misprint for 161 1.

Libraries, Adv. Ed.; Stadt. Breslau'; Stadt. Frankfurt; Min. Schafthausen;

Entdeckung aller Geheimniissen in der | Apocalypsi oder Offenbarung S. Jo- 1 hannis begriffen. | Darinen

[Same as preceding.]

Gedruckt zu Gera, durch Martinum Spiessen | Im Jahr 161 2. |

This impression is identical in every particular with the foregoing, except that in the last line of the Title-page the date 1612 is substituted for 161 1.

Libraries, Stadt Breslau ; Un. Breslau ; Stadt. Zurich ;

Johannis Napeiri, | Herren zu Merchiston, | Eines trefflichen Schottlandischen | Theologi, schone vnd lang gewiinschte | Auszlegung der | Offenbarung Jo- 1 hannis, | In welcher erstlich etliche Propositiones | gesetzt werden, die zu Erforschung desz wahrenVer- 1 stands nothwendig sind : Demnach auch der gantze Text I durch die Historien vnd Geschichten der Zeit erklart, vnnd I angezeigt wirdt, wie alle Weissagungen bisz daher ( seyen erfuUt worden, vnd noch in das kiinfftig | erfullt werden sollen. (

Ausz

Catalogue. 127

Ausz begird der Warheit, vnd der offnung jrer Ge- 1 heimnussen, nach den Frantzosischen, Englischen vnnd | Schottischen Exem- plaren, dritter Edition jetzund auch|vnserem geliebten Teutschen Verstand | vbergeben. | Getruckt zu Franckfort am|Mayn, im Jahr 161 5. |

[Printed in red and black.]

8". Size 6| X 4 inches. Black letter. Xi\ Title. X^\ blank. X'2>-X^\ I4 pages. The Preface ^^ Den Gestrengen^ Edlen, Ehrenvesten, Hochgelehrtetiy Frommen^ Fur- nemmen, Fursichttgen, Ersamen vnd weisen Herrn"; then follow the names, &c., of 2 *^ Biirgernieisterny'' the ^' Statthaliern,'' 2 '^ Seckelmeisterny" the ^'gewesenen Land- vogt zu Louis y " and the " Stattschreibernt 6^r. Sampt einem gantzen Ersamen Rath^ Ibblicher Statt Schaff/iausen, meinen gnddigen vnd gonstigen Herm"^ signed ''Basel den I Augustiy Anno 1615. Ew. Gn. Vnderthdniger, dienstgeflissener Wolffgang Mayer, H. S. D. Dieneram Wort Gottes daselbsten.'" First Table wanting. Ai^-H6S pp. I -123, '' AuszlegMtg vnd Erkldrung der Offenbarung Johannis. Der erste Theil vnd Eyngang dieses Wercks, begreiffend ein ersuchung desz wahren Verstandts der Offenbarung ausz den Leuchtem, gewissen vnd bekanten Puncten, die vngewissere vnnd vnbekante Stiick schlieszlich beweisende, bisz zu vollkommener Erkldrung aller fiimemb- sten Puncten, ingewisse Propositiones abgetheilt.^\ 36 propositions and "Beschlusz " or Conclusion. H62, [p. 124] is blank. On a folding sheet after H62 is " Tabul^ Erkldr- ung vnd Abtheylung der gantzen Offenbarung S. Johannis.^'' H7^-Kk8^ pp. 125-528, ^^ Der ander vndfUmembste Theyl, darinn {nach den hievorgesetzten Fundamenten vnnd GrUnden) die gantze Offenbarung erkldrt vnnd auszgelegt, vnd mit der Historien vnnd Geschicht der Zeit, wie sick die Sachen auff einander verlauffen imd zugetragen, conferiert vnd verglichen wirdt^ mit angehengter Verzeichnusz vnd Erkldrung vber die Orth vnd Spriich so schwerlich zu verstehen, vnd kurtzen Argumentis vnd Lnnhalt eines jeden Capitels.^\' the chapters commence with ''Argument oder Lnnhalt", then follow in three parallel columns '*■ Auszlegung desz Texts", the "Jahr Christi" and *' LListorische Erkldrung", (the 2d and 3d columns are wanting in the chapters men- tioned in the Edin. 1593 edition) ; at the end of each chapter are added " Ferrnere Auszlegung vnd Erkldrung der bezeichneten Oerter dieses Capituls" Lli^-Nn2^ 36 pages, ' * Register Aller denckwicrdigen Sachen, so in diesem Buck hegrieffen, nach dem Alphabet ordentlich zufinden, auszgetheilet"

Signatures. X and A to Z and Aa to Mm in eights + Nn in two =290 leaves.

Paging. 16 + 528 numbered -1-36 = 580 pages.

Errors in Paging. There are several, sheet X especially being in great confusion, but none of the errors affect the last page.

Following H62 on a folding sheet is the second Table.

This edition contains a translation by Wolffgang Mayer of the two Treatises, but without the Text in the second. All the other matter in the English editions is omitted. The additional matter consists of the Preface and the Alphabetical Index to the principal subjects referred to

Q 4 i"

128 Catalogue.

in the book. The first Table is wanting in the copies both of this edi- tion and that of 1627 in all the libraries noted.

Libraries. Kon. Berlin ; Stadt. Breslau ; Un. Breslau ; Stadt. Frankfurt ;

Johannis Napeiri, | Herren zu Merchiston, | Eines trefflichen Schottlandischen I Theologi, schone vnd lang gewiinschte | Auszle- gung der | Offenbarung Jo- 1 hannis. | In welcher erstlich etliche Propositiones | gesetzt warden, die zu Erforschung desz wahren Ver-| standts nothwendig sind : Demnach auch der gantze Text| durch die Historien vnd Geschichten der Zeit erklart, vnd | ange- zeigt wirdt, wie alle Weissagungen bisz daher | seyen erfiillet worden, vnd noch in das kUnff-|tig erfiillt werden soUen. |Ausz begierdt der Warheit, vnnd der offnung ihrer | Geheymniissen, nach dem Frantzosischen, Englischen | vnnd Schottischen Exem- plaren, dritter Edition, jetzund| auch vnserm geliebten Teutschen Ver- 1 standt vbergeben. |

Getruckt zu Franckfurt am | Mayn, Im Jahr 1627. |

[Printed entirely in black.]

8°. Size 6| X 4 inches. Black letter. The type has been reset for this edition, and there are many differences in spelling. The collation, however, is the same as in the edition of 1615, to the end of the Second Treatise, after which we have Lli^-Nn32, 38 \i^(i.%,^^ Register Alter denckwiirdigen Sachen^ . . ." Nn 4, blank.

Signatures. X and A to Z and Aa to Mm in eights, +Nn in four =292 leaves.

Paging. 16 + 528 numbered + 38 + 2 = 584 pages.

Errors in Paging. Rather more numerous than in 1615, sheet X again in confusion, but, as before, the errors do not affect the last page.

The second Table, on a folding sheet, follows H62 as in the 161 5 edition.

Libraries, Adv. Ed. ; Stadt Breslau ; Stadt. Zurich ;

II.

Catalogue. i 29

II. — De Arte Logistica, in Latin.

De Arte Logistica |Joannis Naperi | Merchistonii Baronis|Libn Qui Supersunt. I

Impressum Edinburgi | M.DCCC.XXXIX. | (

4°. Large paper. Size 11 J x 8^ inches. There are 4 leaves at the beginning. The first is entirely blank, on the recto of the second is the single line **/?<? Arte Logistica,"", on the recto of the third is the Title-page as above, and on the recto of the fourth is the Dedication to Francis Lord Napier of Merchiston. The number and arrangement of these pages is slightly different in the Club copies— see note.

On the recto of ai is the word ^^Introduction'' a2^-m32, pp. iii-xciv, ^^Intro- duction.'" by Mark Napier, dated i November 1839.

On the recto of m4 is the line ^^ De Arte Logistica.'\ and on the recto of Ai is the title " The \ Baron Of Merchiston \ His Booke Of Arithmetic ke \ And Algebra. \ For Mr Ilenrie Briggs \ Professor Of Geometrie \ At Oxforde. \ " A2I-D12, pp. 3-26, " Liber Primus. De Computationibus Quantitatum Omnibus Logistica Speciebus Com- munium." D2^-Li\ pp. 27-81, *^ Liber Secundus. De Logistica Arithmetica,''' Li^ blank. L2I-L42, pp. 83-88, ''Liber Tertius. De Logistica Geometrica.'" On the recto of M I is the title " Algebra Joannis Naperi \ Merchistonii Baronis. \ " M2^-P2i, ^^. f)l-ll$, '' Liber Primus. De Nominata Algebres Parte." P22, blank. P3^-Xi^ pp. 1 17-162, *' Liber Secundus. De Positiva Sive Cossica Algebrce Parte." X2, blank.

Signatures. 4 leaves [see notes] + a to m and A to U in fours -f-X in two =134 leaves.

Paging. 8 + xciv numbered + 2+162 numbered + 2 = 268 pages.

There are also two plates, the one a portrait of Napier, the other a view of Mer- chiston Castle.

The collation is from a large-paper copy. Each page is enclosed in a double red line, the title-page being in part printed in red as well as the headings of chapters etc., throughout the work. Generally, how- ever, the copies are printed entirely in black and are without the double line enclosing the type.

In his preface Mark Napier states that he was induced to publish the work *' by the spirited interposition of the Bannatyne and Mait- land Clubs of Scotland." The copies for members of these Clubs are printed entirely in black on their own water-marked paper, size

R loi

130 Catalogue.

loj X 8J inches. They have not a blank leaf at the beginning, but have on the recto of a leaf after the title-page an extract from the minutes authorising the printing, followed by two leaves containing a list of the members of the Club ; the Bannatyne Club having one hundred mem- bers and the Maitland Club ninety. The foregoing differences make the preliminary leaves six instead of four as in the collation.

The manuscript from which the work was published appears, from the following passage in the Memoirs (pp. 419, 420), to be the only one of Napier's papers which survives.

Napier left a mass of papers, including his mathematical treatises and notes, all of which came into the possession of Robert as his father's literary ex- ecutor. When the house of Napier of Culcreugh was burnt, these papers per- ished, with only two exceptions that I have been able to discover. The one is the manuscript treatise on Alchemy by Robert Napier himself; but the other is a far more valuable manuscript, being entitled, " The Baron of Mer- chiston, his booke of Arithmeticke, and Algebra j for Mr Henrie Briggs, Pro- fessor of Geometric at Oxforde. " it is of great length, beautifully

written in the hand of his son, who mentions the fact, that it is copied from such of his father's notes as the transcriber considered "orderlie sett doun."

The treatise on Alchemy is elsewhere stated (pp. 236, 237) to be contained in a thin quarto volume closely written in the autograph of Robert Napier, bearing the title " Mysterii aurei velleris Revelatio ; seu analysis philosophica qua nucleus vera intentionis her mettccB poster is Deum timentibus manifestatur. Authore R. N" and the motto —

*' Orbis quicquid opu?n, vel habet medicina salutis^ Omne Leo Geminis stippeditare potest^

In this connection the following entry may be mentioned which occurs in the sale catalogue of the first portion of the library of the late David Laing : — ** Lambye {J. B.) Revelation of the secret Spirit {Alchymie) translated by R. JV. E. {Robert Napier Edinburgensisf) 1623." The work sold for j^i^ 2s. 6d. Libraries. Adv. Ed.; etc.

1 1 1. — Rabdologiae.

Catalogue. i 3 1

I II. — Rabdologiae. I. Editions in Latin.

Edinbvrgi, I Excudebat Andreas Hart, i6iy.\

12". Size 6| X 3i inches. ^i\ Title. ITi^, blank. 112^-^. 5 pages, '' Illus- trissimo Viro Alexandra Setonio Fermelittoduni Comiti, Fyvoei^ b' Vrqvharti Domino, dfc. Supremo Regni ScoticB Cancellario. S.^\ signed ^ ^Joannes Neperus Merchistonii Baro."* %^, Verses, viz.:— " AvihoH Dignissimo.'\ 4 lines, unsigned; ^* Lectori Rahdologia'\ 4 lines, signed ^^ Patricivs Sand^s''^ \ and ^^ Ad Lectorem.'\ 6 lines, signed "Andreas Ivnivs.'^ 115^-116^ 3 pages, " Elenchvs Capitvm, et vswm totivs operis.^^ 116^ two lines in centre of page. Al^-Bg^, pp. 1-42, " Rabdologia Liber Primvs De usu Virgvlarvm numeratricium in genere." Bio^-Dq^, pp. 43-90,

" RabdologicB Liber Secvndvs De usu Virgularum Numeratricium in Geometricis &* Mechanicis officio Tabularum.'' Dio^-ES^, pp. 91 -112, " De Expeditissimo Mul- tiplicationis Promptvario Appendix. '\ the ♦♦Prgefatio" occupying the first page. E9^-G5^ pp. 1 1 3- 1 54, *^ Arithmetics Localis, qucB in Scacchia abaco exercetur^ Liber unus.'\ the ** Praefatio " occupying the first two pages. G6 blank.

Signatures. U in six + A to F in twelves + G in six = 84 leaves.

Paging. 1 2 + 1 54 numbered + 2 = 1 68 pages.

There are 4 folding plates to face pages loi, 105, 106, and 130, which, with those on pages 6, 7, 8, 94, and 95, are copperplate.

Errors in Paging. None.

In one copy belonging to the Edinburgh University Library the signature B5 is printed in error A5, but in their other copy it is correct.

The word expeditissimo on the 5th and 6th lines of the title-page is in some copies correctly printed expeditis- 1 simo.

Libraries. Adv. Ed. ; Sig. Ed. ; Un. Ed. (2) ; Un. Gl. (2) ; Brit. Mus. Lon.; Un. Col. Lon. ; Roy. Soc. Lon. ; Bodl. Oxf. (5); Un. Camb. ; Trin. Col. Camb. ; Trin. Col. Dub. ; Kon. Berlin ; Stadt. Breslau ; Un. Breslau ; Kon. Off. Dresden ; Un. Halle ; Un. Leiden ; K. Hof u. Staats. Munchen ; Astor, New York ; Nat. Paris ; Un. Utrecht ;

R 2 Rabdologiae

132 Catalogue.

Lvgdvni.|Typis Petri Rammasenij. | M. DC. XXVL|

12°. Size 6i X 3J inches. ti\ Title. +12, blank. \2S-W^ 6 pages, Dedi- cation to Alex. Seton, Lord Dunfermline. +5^, Verses. +5^-t62, ^ pages, Elenchvs Capitvm, with the 2 lines at end. A^-Bg^, pp. 1-42, Rabdologice^ Lib. I. B10I-D62, pp. 43-84, Lib. II. D7I-E32, pp. 85-102, Mult, promptuario. £4^- G4\ pp. 103-139, Arith. Localis. G42-G62, 5 pages, all blank.

Signatures, f in six-h A to E in twelves -t-F and G in sixes =78 leaves.

Paging. 1 2 + 1 39 numbered + 5 = 156 pages.

There are 9 folding diagrams to face pages 49, 51, 59, 81, 94, 97, 98, 115, and 117 ; those facing pages 94, 97, 98, and 117, correspond to the 4 folding plates of the 1617 edition, the others are tables which in 1617 were printed in the text.

The numbering of the pages, though somewhat indistinct, seems to be correct throughout. In printing the signatures, however, C7 is numbered in error C6, and E3 has no signature printed.

This edition, published at Leyden, contains exactly the same matter as that of the Edinburgh edition of 1617. None of the plates, however, are engraved on copper. The decimal fractions are printed according to Simon Stevin's notation; thus, for example, on p. 41 we have 1994 Q, 9 0 I 0 ^ 0 ° Cl) ^^^^^ ^^ ^'^^ ^^^7 edition it is printed

I II III nil

1994,9 160.

Libraries. Un. Ed.; Un. Ab. ; Un. St And.; Greenock; Bodl. Oxf. ; Chetham's Manch.; Trin. Col. Dub.; Kon. Berlin; Un. Breslau ; Stadt. Frankfurt ; K. Hof u. Staats. Miinchen ; Astor, New York ;

Lvgd. Batavorvm. | Typis Petri Rammasenij. | M. DC. XXVIIL I

This

Catalogue. 133

This edition is identical with that of 1626, described in the previous entry, but the original title-page has been cut out, and the above sub- stituted. The only important change in this new title-page, besides the alteration of date, is the substitution of the name Lvgd. Batavorvm for LVGDVNI, and the object in printing a new title-page was probably to effect this change in name, as confusion may have arisen from the single word * Lugduni ' being used for Leyden, instead of the more common form Lugd. Batavorum, the word Lugdunum being the usual Latin form of Lyons, as, for example, in the 1620 edition of the ' Descriptio.'

Libraries, Adv. Ed. ; K. Hof u. Staats. Miinchen ; Nat. Paris :

2. Edition in Italian.

In Verona, Appresso Angelo Tamo. 1623. | Con licenza de' Superiori.|

8°. Size 6 J X 4| inches. ^\^, Title. ti^, blank. +2i-t3i, 3 pages, ".^//^

Illmo. ^ Ecc^no. Sigre. Teodoro Triwltio, Prencipe del Sac. Rom. Imperio^ di Musocco, &> della Valle Misolcina ; Conte di Melzo, d^ di Gorgonzola ; Signer di Codogno, dr» di Venzaghello ; Caualier delV Ordine di S. Giacomo^ &"€.", dated and signed "Z>2 Verona li 12. Febraio 1623. . . . Marco Locatelli." +3', *' Al medesimo Sig. Prencipe Triwltio LHstesso Locatelli. ", followed by 10 lines of verse. t4S ''Del Sig. Ambrosio Bianchi Co. Cau. e I. C. Coll. di Mil. Al Sig. Can. Marco Locatelli.'\ with 14 lines of verse. t4^ "L>el Sig. Francesco Pona Med. Fis. &> Ace. Filarm. Al medes. Sig. Cau. Locatelli.'\ with 13 lines of verse.

f^i-fgi, 7 pages, ''Racconio De' Capi di tutta V Opera, Et de'Titoli piu rileuanti in essi. " On +82 is printed ''Imprimatur Fr. Siluester Inquisitor Verona:. Augustinus Dulcius SeremssimcB Reip. Veneta Seer.'' Ai^-Y/[\ pp. 1-95, "Della Raddologia

J^ -2 Libro

134 Catalogue.

Libro Prima. DelT vso delle Virgole numeratrici in genere. " F42, blank, F5^- K4^, pp. 97-159, ^^ Delia Raddologia Libro Secondo. DelPvso delle Virgole numeratrici nelle cose Geometriche^ ^ Mecaniche, con taiuto di alcune Tauole." K42, blank. K5I-N32, pp. 161-210, ^^ Prontvario Ispeditissimo Delia Molteplicationey^ the **Pro- emio " occupying the first 2 pages. N4^-Q8^, pp. 211-269, ^' Arimmetica Locale^ Che nel Piano dello Scacchiere si esercita. ", the " Prefatione " occupying the first 2 pages. On QS'-^, * * // Fiiu di tutta V Opera. In Verona^ Appresso Angelo Tamo, 1623. Con licenza de' Superiori."

Signatures. + and A to Q in eights =136 leaves + 7 diagrams interleaved and in- cluded in paging = 143 leaves.

Paging. 1 6 + 269 numbered + 1 = 286 pages.

There are 7 diagrams on interleaved and folded sheets, each of which counts as two pages ; the sides containing the diagrams are numbered as pages 25, 36, 49, ^T^y 169, 179, and 233.

Errors in Paging. P. 75 not numbered, and pp. 251, 266, and 267 numbered in error 152, 264, and 165.

New Dedication and Complimentary Verses are substituted for those in the edition of 16 17, and there are numerous notes throughout the work by the Translator, as well as additions and alterations. One of these may be mentioned. At the end of the work Napier adds these words, "Atque hie finem Arithmetics Locali imponimus. DEO soli laus omnis & honor tribuatur. FINIS.", but his Italian translator makes the champion of Protestantism say, " Con che a questa nostra ARIMMETICA LOCALE poniamo fine, a DIO, & alia Beatissima Vergine MARIA tutta la gloria, & I'honore attribuendo. Amen."

Of the four folding diagrams in the edition of 161 7, the two facing pages, loi and 130, are represented by the diagrams at pages 179 and 233, but the other two are not given in this edition.

Libraries. Un. Ed.; Brit. Mus. Lon. ; Un. Col. Lon.; Trin. Col. Camb. ; Nat. Paris ;

3. Edition in Dutch.

Waer achter bygevoegt zijn eenige seer lichte manieren van

Rekenen

Catalogue. 135

Noch is hier achter byghevoeght de Thiende van | Symon Stevin van Brugghe.|

Ter Govde, | By Pieter Rammaseyn, Boeck-verkooper inde corte 1 Groenendal, int Vergult ABC. 1626. | Met Previlegie voor thien laren. |

4". Size 8| X 6f inches. * i^ Title-page. ^\\ ♦' Copie Van De Pre-

vilegie" granted by the States-General to Adrian Vlack for ten years, signed at s'Gra- venhaghe, 24 Dec. 1625. *2, 2 pages, The dedication, '* Toeeyghen- brief

Aende Doorlvchtighe, Hooge Ende Mogende Heeren^ niijn Heeren de Staten Generael vande Vereenighde Nederlanden. Mitsgaders De Edele, Emtfeste Ende Wyse Heeren^ de Heeren Gecommitteerde Raden van Hollant ende Westvrieslant. Als Mede Aende Ackt- bare, Voorsienige Heeren, mijn Heeren Bailiu, Burghemeesteren, Schepenen, ende Vroet- schap der Vermaerde Stadt Gotida." signed by Ezechiel de Decker at Gouda 4 Sep. 1626. * 3^-* 4S 3 pages, the preface, ^^Voor-reden tot den Goetwilligen ende Konst- lievenden Leser" signed by Ezechiel de Decker at Gouda 4 Sep. 1626.

* 4^, Three Latin verses : " loanni Nepero\Avthore Dignissimo. |," 4 lines ; ^^ Lectori KabdologicB. \ ", signed Patricius Sandaeus, 4 lines ; and "Ad Lectorem. \ ", signed Andreas Junius, 6 lines. ti^-+2^ 3 pages, The index, ^'Register van alle de Hooftstucken,

ende Ghebruycken deses gantschen Boeckx.'^ fz^ ''De Druck-fauten salmen aldus verbeteren.^\ 2i lines of errata. Ai^-E42, pp. 1-40, "loatmis Neperi Eerste

Boeck, Vande Tellingh door Roetjes. Van Het Ghebrvyck Der Telroetjes int ghemeen." . in nine chapters. Fi^-L4\ pp. 41-87, " loanni s Neperi Tweede Boeck, Vande

Tellingh door Roetjes. Van Het Ghebrvyck Der Tel-Roeties in Meetdaden, ende Werckdaden, met behulp van Tafels. ", in eight chapters. L42, blank.

M1I-O42, pp. [89H112], " loannis Neperi Aenhanghsel Van Het Veerdigh-Ghereet- schap van MenighwldiginghP, in four chapters, the title is on p. [89] and the Pre- face on p. [90], the last page, O42, being blank.

P1I-T22, pp. [ii3]-i48, "Joannes Nepervs Van de Plaetselicke Telkunst:\ in eleven chapters, the title is on p. [113] and the Preface on p. [ii4].

Vi^-Rr 42, pp. [i49]-3o8, "Ezechiel De Decker Van Coopmans Rekmingen. Leerende Door Thiendeelighe Voortgangh sonder gebrokens met wonderlicke lichticheyt afveerdigen alle ghemeene Rekeninghen. '\ in eight chapters, the title is on p. [149] and the Preface on p. [150].

ai^-q4^ 128 pages. Tables.

De I Thiende. | Leerende Door | onghehoorde lichticheyt alle

R 4 re-

136 Catalogue.

re- 1 keninghen onder den Menschen noodigh val- 1 lende, afveer- dighen door heele ghetal- 1 len, sender ghebrokenen. | Door Simon Stevin van Brugghe. |

Ter Govde, | By Pieter Rammaseyn, Boeck- 1 vercooper, inde Corte Groenendal, int Duyts | Vergult ABC. | M. DC. XXVI. |

Ai^ Title-page. Ai^, blank, A2^-A3\ pp. 3-5, Preface ''Den Sterrekiickers, Landtmeters, Tapiitmeters, Wijnmeters, Lichaemmeters int ghemeene, Muntmeesters, ende alien Cooplieden, wenscht Simon Stevin Gheluck.^'' A32, p. 6, ** Cort Begriip." A4, pp. 7 and 8, '' Het Eerste Deel Der Thiende Vande Bepalinghen." B1I-B4I, pp. 9-15, '* Het Ander Deel Der Thiende Vatide IVercl'ing/ie." 64^-02^, pp. 16- 27, Aenhanghsel. D2^, blank.

Signatures. * in four and f in two ( = 6) + A to Z and Aa to Rr in fours, except T, Y, and Cc, which are in twos, (=154) + a to q in fours ( = 64) + Ato C in fours and D in two ( = 14) = in all 238 leaves.

Paging. 12 + 308 numbered + 1 28 + 27 numbered + 1 = 476 pages.

Errors in Paging. The pages III, I2i, 218, 219, 270, 271, 274, are numbered no, 221, 217, 218, 254, 255, 258, respectively, and the numbers are not printed on the following pages, 88-90, 103, 105, 112-114, 149, 150, 169-176, 187, 196, 222, 275-284, but the numbering of the last page, 308, is not affected.

Errors in Signatures. N3 is printed as N5, V2 as V, and Gg2 as Gg3.

The leaf K4 has been cut out and another substituted.

The translation of Rabdologise, extending from -^42 to T22 and em- bracing 5 unnumbered and 148 numbered pages, appears to correspond exactly with the original Latin edition of 1617, except that Napier's dedication to Lord Dunfermline on the 5 pages 1F2i-1l4^ and the two lines on the page 1162 Qf ^\^2X edition are omitted. The translation is by Adrian Vlack, and was made at the request of De Decker, for this work.

Libraries. Un. Col. Lon. ; Trin. Col. Camb. ; Kon. Berlin ; Kon. Hague ; Nat. Paris ;

IV.— Mirifici

Catalogue. 137

IV. — Mirifici logarithmorum canonis descriptio

and, Mirifici logarithmorum canonis constructio.

I. Editions in Latin,

Edinbvrgi, | Ex ofificina Andreae Hart | Bibliopdlae, do. dc.

XIV. I

[The title is enclosed in an ornamental border. A reproduction of the Title-page will be found at p. 374 of the Memoirs.]

4°. Size 7i X 6| inches. Ai^, Title. Ai'-', blank. A2, 2 pages, ^^ Jllicstrissimoy ^ optima spd Principi Carolo, Potentissimi, (Sr" Invictissimi^ lacobi D. G. magna Britannia^ Francice, dr' HibernicE Regis^ filio tinico, Wallice Principi^ Duct Eboraci, dr" RothesaicBy magno Scotice Senescallo, ac Insularum Domino, &"€. D. D. Z>.", signed " Joannes Nepervs. " A3^, " Jn Mirificvm Logarithmorum Canonem Prcefatio. " h'^-K^y 2 pages. Verses : — ^^ Ad Lectorem Trigonometries studiosum."^ 12 lines signed " Patricius Sand^us." ; '*/« Logarithmos D, I. Neperi'\ 10 lines ; ^* Aliud.^\ 6 lines; "Ad Lectorem.'", 4 lines signed ^'Andreas Lvnivs Philosophice Professor in Academia Edinburgenay A42, " Ln Logarithmos." , /^Wnts. Bi^-D2'', pp. I-20,

" Mirifici Logarithmorum canonis descriptio, eiusque usus in utrdque Trigonometria, ut etiam in omni Logistica fnathematica, amplissimi, facillimi, dr* expeditissimi explicatio. Liber J. ''^ T>^-\\^, pp. 21-57, '' Liber Secvndvs, De canonis mirifici Logarith-

morum prceclaro usu in Trigonometria.'', on p. 57 after the ^^ Conclvsio." follow *^ Errata ante lectionem emendanda.", 7 lines ; and the last line of the page is *'5«- quitur Tabula seu canon Logarithmorum, '' li^ and ai^-mi^, 90 pages, The Table. mi^, " Admonitio" or blank [see note].

Signatures. A to H in fours + 1 in one -fa to 1 in fours + m in one = 78 leaves.

Paging. 8 + 57 numbered -1-91 = 156 pages.

Errors in Paging. In some copies pp. 14 and 15 are numbered 22 and 23 [see note].

S Table

138

Catalogue.

Gr.

mtn\

30

Sinus

Logarithmi

+

Differentice

'ogarithmi

Sinus

5000000 5002519 5005038

6931469 6926432 6921399

5493059 5486342 5479628

438410 440090 441771

8660254 8658799 8657344

5007556 5010074 50T2591

6916369 6911342 6906319

5472916 5466206 5459498

443453 445136 446821

8655888

8654431 8652973

5015108 5017624 5020140

6901299 6896282 6891269

5452792 5446088

5439387

448507 450194 451882

8651514 8650055 8648595

9

10

II

5022656 5025171 5027686

6886259 6881253 6876250

5432688

5425992 5419298

453571 455261 456952

8647134

8645673 86442 II

12 13 14

5030200 5032714 5035227

6871250 6866254 6861261

5412605

5405915 5399227

458645 460339 462034

8642748 8641284 8639820

15 16

18 19

5037740 5040253 5042765

6856271 6851285 6846302

5392541 5385858 5379177

463730 465427 467125

8638355 8636889

8635423

5045277 5047788 5050299

6841323 6836347 6831374

5372499 5365822

5359147

468824

470525 472227

8633956 8632488 8631019

21 22 23

5052809

5055319 5057829

6826405 6821439 6816476

5352475 5345805

5339137

473930 475634 477339

8629549 8628079 8626608

24

25 26

5060338 5062847 5065355

6811516 6806560 6801607

5332471 5325808

5319147

479045 480752 482460

8625137 8623665 8622192

27 28 29

5067863 5070370 5072877

6796657 6791710 6786767

5312488

5305831 5299177

484169

485879 487590

8620718 8619243 8617768

30

5075384

6781827

5292525

489302

8616292 30

59

There

Catalogue. 139

There are two noticeable varieties of this edition, the one with an Admonitio printed on mi 2, the back of the last page of the table, the other with that page blank. In general the former variety has the error in paging before mentioned, while the latter has the paging correct. There are also, however, copies which want the Admonitio but have the error in paging, for instance, one of the copies in the Bodleian and the copy in University College, London. A translation of the Admonitio referred to above is given in the Notes, page 87.

A specimen page of the table is given opposite, and a full de- scription of its arrangement will be found in section 59 of the Con- stnictio.

Libraries.

(i.) With Admonitio. Sig. Ed. ; Un. Ed. (2) ; Hunt. Mus. Gl. ; Un. Ab. ; Brit. Mus. Lon. (2); Roy. Soc. Lon. ; Bodl. Oxf. (3); Un. Camb. ; Trin. Col. Dub. (2);

(2.) Without Admonitio. Adv. Ed. ; Un. Gl. (2) ; Un. Col. Lon. (see note) ; Bodl. Oxf. (see note) ;

Foreign Libraries^ varieties not distinguished. Kon. Berlin ; Stadt. Bres- lau; Un. Breslau; Stadt. Frankfurt; Pub. Geneve; Un. Halle; Un. Leiden; Un. Leipzig ; K. Hof u. Staats. Miinchen ; Nat. Paris ;

A reprint of the Mirifici Logarithmorum Canonis Descriptio is contained in Scriptores Logarithmici ;|or|A Collection | of | Several Curious Tracts I on the [Nature And Construction! of [Logarithms, | men- tioned in Dr Hutton's Historical Introduction to his New [Edi- tion of Sherwin's Mathematical Tables : | together with [Some Tracts on the Binomial Theorem and other subjects [connected with the Doctrine of Logarithms. [Volume VI. [

London. | Printed by R. Wilks, in Chancery-Lane ; [ and sold by J. White, in Fleet-Street. | MDCCCVII. [

The work, Scriptores Logarithmici, consists of six large quarto vol- umes, and was compiled by Baron Francis Maseres. The volumes appeared in the years 1791, 1791, 1796, 1801, 1804, and 1807 respect-

S 2 ively.

I40 Catalogue.

ively. The reprint, which will be found on pages 475 to 624 of the sixth volume, gives the Descriptio and the Canon in full, with the Admonitio on its last page.

Graesse states that the edition of 16 14 was "Reimpr. sous la m^me date dans les Transact, of the Roy. Soc." He probably refers to this reprint as Baron Maseres was a member of the Royal Society.

Libraries. Adv. Ed. : etc.

Mirifici | Logarithmo- 1 rvm Canonis | Descriptio, | Ejusque usus, in utraque Trigonome-|tria ; vt etiam in omni Logistica Ma-| thematica, amplissimi, facillimi,| & expeditissimi explicatio.| Accesservnt Opera Posthvma;|Prim6, Mirifici ipsius canonis con- structio, & Logarith- 1 morum ad naturales ipsorum numeros habi- tudines. | Secund6, Appendix de alia, eaque praestantiore Loga- 1 rithmorum specie construenda. | Terti6, Propositiones quaedam eminentissimae, ad Trian- 1 gula sphaerica mira facilitate resol- venda. | Autore ac Inventore loanne Nepero, | Barone Mer- chistonii, &c. Scoto.|

Edinbvrgi,! Excvdebat Andreas Hart. | Anno 1619.I

[The ornamental part of the Title-page is the same as in 1614, the type only being altered.]

4°. Size 7^x61 inches. [See note.]

Edinbvrgi, I Excudebat Andreas Hart. | Anno Domini 1619.]

Ai^, Title. Ai2, blank. A2, 2 pages, " Lectori Matheseos Studioso S. ", signed

"Roberivs Nepervs, F." A3^-E4^, pp. 5-39, " Mirifici Logarithmorvni Canonis Con- strvctio; {Qvi £t Tabvla Ai'tificialis ab autore deinceps appellatur) eortcmque ad natu- rales ipsorum numeros habitudines." , Y.^-Y'^, pp. 40-45, ^^ Appendix^* containing ** De alia eaque prcsstantiore Logarithmorvm specie construenda; in qua scilicet, vni-

talis

Catalogue. 141

tatis Logarithmus est o." ; ^^ Alius modus faciU creandi Logarithmos numerortim com- positorum, ex dcUis Logarithmis suorum primorum. " ; < ' Habitudines Logarithmorvm iSr* suorum naturalium numerorum invicem." F3^-G3^, pp. 46-53, "LvcvbrcUiones

Aliqvot Doctissirni D. Henrici Briggii In Appendicem pramissam." G3^-H3'', pp.

54-62, " Propositiones Qvcedam Eminentissimce ad triangula sphcerica, mird facilitate resolvenda." coniainmg "Triangulum sphcericum resolvere, absque eiusdem divisione in duo quadrantalia aut rectangula. " ; " De semi-sinuum versorum prcestantia <S^• z;j«." H4^-l2^, pp. 63-67, " Annotationes Aliqvot Doctissimi D, Henrici Briggii In Propo- sitiones Prcemissas. " l2^, blank.

Signatures, A to H in fours + 1 in two =34 leaves. Paging. 67 numbered + 1 = 68 pages.

The first title-page, given above, with a blank leaf attached, appears to have been printed in order that it might be substituted for the title- page of the 1614 edition of the Descriptio by those who desired to have the two works on logarithms bound together. In such cases the 16 14 title-page is usually cut out and the new one pasted on in its place. Consequently, in these copies, we find the same varieties as mentioned in the preceding entry. In other copies, however, only the new title- page and blank leaf are inserted before the Constructio.

Libraries.

I. Copies containing both the Descriptio and Constructio, with the new title-page substituted.

1. With Admonitio, Un. Ed.;

2. Without Admonitio. Adv. Ed. ; Sig. Ed. ;

II. Copies containing the Constructio only with the new title-page and blank leaf attached. Un. Ed.; Un. Col. Lon.; Bodl. Oxf. ; Un. Camb. ; Trin. Col. Dub. ;

Foreign Libraries, varieties not distinguished. Un. Halle;

Lvgdvni, | Apud Barth. Vincentium. | M. DC. X X. | Cum Priui- legio Caesar. Majest. & Christ. Galliarum Regis. |

[Printed in black and red.]

S3 ®'

142 Catalogue.

8°, printed as 4°. Size 8^x5^ inches. Ai^ Title. Ai^ blank. A2, 2 pages, Dedication to Prince Charles [see note]. A3^ Preface. A3^-A42, 3 pages, Verses. B1I-D22, pp. 1-20 The Description Lib I. D3I-H42, pp. 21-56, Lib. II.

Sigfiatures. A to H in fours = 32 leaves.

Paging. 8 + 56 numbered = 64 pages.

Errors in Paging. None, but sig. D3 is printed D5.

Logarithmes. | A Scavoir, | La Table de I'admirable inuention pour I promptement & facilement Abreger les sup- 1 putations, d'Arithmetique auec son vsage, en rv|ne & I'autre Trigonometrie, & aussi en toute | Logistique Mathematique. |

Lvgdvni, | Apud Barthol. Vincentivm. | Cum priuilegio Caesareo & Galliarum Regis. |

Ki^ Title. Ki'^-Uz^, Table. Ma^, * ^;r/ra?V/ ' or blank [see note]. Signatures. A to L in fours + M in two =46 leaves. Paging. 92 pages not numbered.

Ai^ Title. Ai^ blank. A2, pp. 3 & 4, ^^ Roberivs Nepervs Avctoris Filivs Lectori Matheseos Studioso. .S." A3^-E2^, pp. 5-35, The Constructio. E22-Fi\ pp. 36-41, The Appendix. Fi^-Gi^, pp. 42-49, Lvcvbrationes by Briggs [see note]. Gi^-Hi^, pp. 50-57, Propositiottes Trigonornetricce. H12-H32, pp. 58-62, Annotationes by Briggs. H4^, ' Extraict ' or blank [see note]. H4'* blank.

Signatures. A to H in fours = 32 leaves.

Paging. 62 numbered + 2 = 64 pages.

Errors in Paging. None.

On the issue of the Edinburgh edition of 16 19, Barth. Vincent would

appear

Catalogue. 143

appear to have at once set about the preparation of an edition for issue at Lyons, and, as will be seen from the next entry, had some copies printed with the date 1619 on the first title-page. The three parts are usually found together, but some copies contain only the Descriptio and Tabula. The Admonitio is omitted from the last page (M22) of the Tabula, but in many copies its place is taken by the "Extraict du Priuilege du Roy," at the end of which is printed " Acheue (Tlmprimer le premier Octobre, mil six cents dixneuf." The copies in the Advocates* Library, Edinburgh, and Astor Library, New York, have this Extraict on M22 of the Tabula, and have also on H4^ of the Constructio the Extraict reset with the note at end altered to ^^ Mirifici Logarithmorum Acheu'e dHmprimer /(? 31 Mars 1620."

The edition is a fairly correct reprint of the Edinburgh one, but the decimal notation employed by Briggs in his Remarks on the Appen- dix has not been understood, the line placed by him under the frac- tional part of a number to distinguish it from the integral part being here printed under the whole number. The only intentional alteration, besides the title-page, is in the Dedication to Prince Charles, where " Francise " is omitted from his father's title, " magna Britannia, FrancicBj d^ HibernicB Regis."

Libraries. Adv. Ed.; Un. Ed.; Act. Ed.; Un. Gl. ; Un. St. And.; Brit. Mus. Lon. (parts i and 2 only); Un. Col. Lon.; Roy. Soc. Lon.; Kon. Berlin ; Un. Breslau (parts i and 2 only) ; Kon. Off. Dresden ; K. Hof u. Staats. Miinchen ; Astor, New York; Nat. Paris; Un. Utrecht; Stadt. Zurich (parts i and 2 only) ;

[Same as preceding.]

Lvgdvni, | Apud Barth. Vincentium. | M. DC. XI X. | Cum Priui- legio Caesar. Majest. & Christ. Galliarum Regis. |

[Printed in black and red.]

The only respect in which this entry differs from the preceding is in the date on the title-page. A possible explanation of this may be that the title-page was originally set up with the date m. dc. xix., but

S 4. when

144 Catalogue.

when it was found that the whole work could not be issued in that year, the date was altered to m. dc. x x., and a few copies may have been printed before the alteration. The only copy which we have found is in the Bibliothbque Nationale. The volume contains the three parts ; the Tabula has M42 blank ; the Constructio has on its title-page the usual date of 1620, and has on H4^ the Extraict the same as in the Advo- cates' Library copy mentioned in the preceding entry. Library. Nat. Paris;

Lvgdvni, | Apud loan. Anton. Hvgvetan, | & Marc. Ant. Ravavd. |m. DC. LVlii. |

[Printed in black and red.]

This issue is evidently not a new edition, but the remainder of the edition of 1620 with the following alterations. In the Descriptio sig- nature A has been reprinted with title-page as above, and several other less important alterations. The Tabula is unaltered, still retaining the name of Barth. Vincent on the title-page. The Constructio has the first two leaves cut out so that the first page is numbered 5. The Extraict is often wanting on M22 of the Tabula, but in the copies examined is printed on H4^ of the Constructio, exactly as in the Advocates* Library copy of 1620, the name of the work in the Extraict being that on the first title-page of the 1620 edition, and not that used in the title-page given above.

Libraries. Adv. Ed. ; Un. Gl. ; Kon. Berlin ; Un. Breslau ; Un. Halle ; Stadt. Ziirich :

2. Editions in English of the Descriptio alone.

vse

Catalogue. 145

London, | Printed by Nicholas Okes. | 1616. |

12°. Size 51 X 3f inches. Ai^ Title. Ai2 blank. Kz^-K-}^, 3 pages, " To The

Right Ho7iovrable And Right Worshipfvll Company Of Merchants of London trading to the East-Indies, Samvel Wright wisheth all prosperitie in this life, and happinesse in the life to come.'' Kf-K\-, 3 pages, " To The Most Noble Atid Hopefvll Prince, Charles: Onely Sonne Of the high and mightie lames by the grace of God, King of great Brittaine, France, and Ireland: Prince of Wales: Duke of Yorke and Rothesay: Great Steruard of Scotland: and Lord of the Islands." signed Wohn Nepair.' A5, 2 pages, ** The Authors Preface to the Admirable Table of Loga-

rithmes,". AS^-AS^, 6 pages, " The Preface To The Reader By Henry Brigges.",

signed * H. Brigges.' A9, 2 pages, Lines, "/« praise of the neuer-too-mtich

praised Worke and Authour the L. of Marchiston.", 54 lines, ^^ By the vnfained louer and admirer of his Art and matchlesse vertue, lohn Dailies of Hereford." Aio cut out in all copies. All, 2 pages, Lines, " In the iust praise of this Booke, Authour, and Translator.", 49 lines, signed ^^ Ri. Leuer." Ai2\ **^ Vieiu Of This

Booke." A122, '■'■ Some faults haue esraped in printing of the Table, . . . .",

58 corrections are given. Bii-C3\ pp. 1-29, "A Description Of The Admir-

able Table Of Logarithmes, With The Most Plentifvl, Easie, And Ready Vse thereof in both kindes of Trigonometrie, as also in all Mathematicall Accounts. The First Booke." C32-E9I, pp. 30-89, *' The Second Booke." Y.(f-l 6^ 90 pages, The

Table. I 6^, blank. After I 6^, on a folding sheet, is an engraved diagram of the ''Triangular instrument all Table." I 7I-K22, pp. 1-8, *' The Vse Of The Tri-

angular Table for the finding of the part Proportionall, penned by Henry Brigges. ", also on p. 8, ''Errata in the Treatise.", 8 corrections.

Signatures. A to H in twelves + 1 in eight + K in two= 106- 1 = 105 leaves, A 10 being cut out. Leaves Eio and Eii have also been cut out, but in their place two new leaves are inserted.

Paging. 22 + 89 numbered + 91+8 numbered = 2io pages. Also plate following 16'^.

Errors in Paging. None.

The Table is to one place less than the Canon of 16 14, but the

"Y logarithms

146 Catalogue.

logarithms of the sines for each minute from 89°-9o° are given in full, the last figure being marked off by a point. This is, I believe, the earliest instance of the decimal point being used in a printed book. The Admonitio at the end of the Table is wanting.

The two words " and maintaine " in the last line of the first page of Briggs' preface (A6I) are ruled out in ink in all copies both in this edition and in that of 16 18.

Libraries. Adv. Ed. ; Un. Gl. ; Brit. Mus. Lon.; Bodl.Oxf.; Qu.Col. Oxf. ;

London,! Printed for Simon Waterson.| 1618. |

This edition is really that of 16 16 with the title-page cut out and the above put in its place ; there being also added at the end of the work (A3^-Aio2, pp. 1-16) ^^ An Appendix to the Logarithmes, shelving the practise of the Calculation of Triangles^ and also a new and ready way for the exact finding out of such lines and Logarithmes as are not precisely to be found in the Canons J^

One of the copies in the Glasgow University Library has the new title-page with blank leaf attached, inside of which is placed the sig. A of the 1 6 16 edition with its first leaf cut out, and also the new sig. A (A3- A 10) containing the Appendix.

Signatures.

Catalogue. 147

Signatures. As in i6i6 edition 105 leaves + A, in eight (commencing with A3 and ending with A 10), = 113 leaves.

Paging. As in 1616 edition 210 pages + 16 numbered = 226.

Libraries. Un. Ed.; Un. Gl. (2); Roy. Soc. Lon. ; Bodl. Oxf.; Un. Camb. ; Trin. Col. Camb. ;

The Wonderful I Canon Of Logarithms | or the | First Table Of Logarithms I with a full description of their ready use and easy| application, both in plane and spherical trigono- 1 metry, as also in all mathematical | calculations. | Invented and published by | John Napier, I Baron of Merchiston, etc., a native of Scotland, A.D. 1614. 1 Re-translated from the Latin text, and enlarged with a table of [hyperbolic logarithms to all numbers from i to I20i.| By Herschell Filipowski.

Published for the Editor] By W. H. Lizars 1 3 St. James' Square, Edinburgh.] 1857. |

16°. Size 6^x3§ inches. ai^ Title, a 1 2, blank. a2i, '' This edition is in- scribed to William Thomas Thomson, Esq " a2^, blank. A3, pp. v and

vi, Dedication to Prince Charles, signed ^' John Nepair.^'' a*, pp. vii. and viii.. The Author's preface. a5^-a62, pp. ix-xii, ** The Preface To The Reader By Henry Briggs.'' z.'j^-hi^ pp. xiii-xviii, *^ Translator s Preface." hi^,^. x\\, Notes.

b22, Errata. Al''-B42, pp. 1-24, Book I. B42-F4I, pp. 24-71, Book II. Y\\ ''Note to Table II. by the Translator.'' Ai^-F62, 92 pages, " Table I., Napier^ s Logarithms of Sines."", the title occupies the first page, the last is blank, and the table occupies the intervening 90. Fy^-GS^ 20 pages, ** Table II., Napier's

Logarithms to Numbers, called also Hyperbolic Logarithms, from i-oi to 1200.", on first page is the title, then follow the table occupying 18 pages, and on the last page is printed ** END " within an ornamental device.

Signatures, [a] in 8 + b in 2, + A, B, C and E in eights + F in four + A to G in eights = 102 leaves.

Paging. XX numbered + 72 numbered + 112 = 204 pages.

The numbers and logarithms in Table I. are those of the Canon of 1 6 14, each divided by 10,000,000, so that the logarithms are strictly to base e-\ The Admonitio at the end of the Table is wanting. The logarithms in Table II. are to base e.

Libraries. Act. Ed. ; Brit. Mus. Lon. ; Act. Lon. (2).

T 2

ipjiiiiu.* wr rii»««J iioriifiri

!LH.X..IX-U' J'l

APPENDIX.

In the preparation of the foregoing Catalogue, several works by other Authors were met with which have considerable interest from their connection with the works of Napier. It seemed desirable to preserve a record of them, and they are accord- ingly given below, with such particulars as were noted at the time.

Napiers I Narration : I Or, I An Epitome | Of | His Booke On The| Revelation. I Wherein are divers Misteries disclosed, | touching the foure Beasts, seven Vials, seven Trumpets,] seven Thunders, and seven Angels, as also a discovery of| Antichrist : together with very probable conjectures [touching the the time of his destruc- tion, and I the end of the World. | A Subject very seasonable for these last Times. |

Revel. 22. 12. 1 And behold I come shortly, and my reward is with me, to give to every man | according as his worke shall be. |

London, | Printed by R. O. and G. D. for Giles Calvert, 1641. |

4". Size inches. AiS Title. Ai^, blank. A2I-C32, 20 pages,

^^ Napier's Narration Or An Epitome of his Booke on the Revelation.'''* C4, 2 pages, blank.

Signatures. A, B, and C in fours =12 leaves.

Paging. 2 + 20 + 2 = 24 pages, not numbered.

Errors in Signatures. C2 numbered in error C3.

This tract is written in the form of a dialogue, wherein Rollock is made the scholar and Napier the master : see Memoirs, p. 175.

Libraries. Brit. Mus. Lon.; Bodl. Oxf.

The

Appendix. 149

The bloody Almanack :|To which England is directed, to fore- know what shall I come to passe, by that famous Astrologer, M. John Booker. I Being a perfect Abstract of the Prophecies proved out of Scripture, | By the noble Napier, Lord of Marchistoun in Scotland. |

London | Printed for Anthony Vincent, and are to be sold in the Old-Baily. 1643. |

[With large woodcut in centre of page containing symbolical designs,]

4°. Size inches. Ai^, Title. Ai^, blank. h2>-k£^, pp. i-6,

" The bloody Altfianack" containing " I. Concerning the opening of the seven Scales mentioned Revel. 6." ; ** II. Concerning the seven Trumpets mentioned chap. 8 & 9." ; "III. Concerning the seven Angels mentioned Rev. 14."; "IIII. Concerning the Symboll of the Sabboth." ; " V. Concerning the Prophesie of Elias." ; " VI. Concern- ing the Prophecie of Daniel." j " VII. Concerning Christ's ovene saying."

Signature. A in 4 = 4 leaves.

Paging. 2 + 6 numbered = 8 pages.

Libraries. Brit. Mus. Lon.;

Another Edition.

The bloody Almanack : | To which England is directed . . . . . . I By the noble Napier, Lord of Marchistoun in Scotland. | With Additions.!

London | Printed for Anthony Vincent, and are to be sold in the Old-Baily. 1643.I

[The printing round the woodcut is slightly altered.]

The additions are on Ai^ ''A Table . . . ."and '' M. J. Booker his Verses . . . .", also on A42 at the end an added Note.

Libraries. Brit Mus. Lon. ;

[With symbolical woodcut surrounded by the signs and names of the zodiac.1

T 3

150 Appendix.

4*. Size inches. Ai\ Title. Ki^-Ki^ 3 pages, Astrological predic-

tion of events **/« Jannary^^ ^^ In February ^^^ etc. A3^-A42, 4 pages, "I.

Concerning the seaven Angels mentioned Rev. 14."; "II. Concerning the Symbol! of the Sabbath." ; " III. Concerning the Prophesy of Elias." ; "IV. Concerning the Prophecie of Daniel."; **V. Concerning Christ's own saying."; these contain the same matter as in the Almanack of 1643, but the first two subjects there treated of are here omitted.

Signature. A in 4=4 leaves.

Paging. 8 pages not numbered.

Libraries. Brit. Mus. Lon.; Bodl. Oxf.;

Heureux sur qui le Soleil d'intelli- |gence se leue. |

louxte la coppie imprimee k Rouen. |

A Paris, | Chez Abraham le Feurq, rue sainct Ger- 1 main de Lauxerrois.|M.C.D.X.|Auec Priuilege du Roy.j

8°. Size 6i X 4 inches. AiS Title. Ai«, blank. A2, pp. 3 & 4. Dedi- cation ** A Tres-havt Et Tres-pvissant Seignevr, Messire Guillaume de Hautemer, sieur de FeruaqueSy .... Sc Baron de Manny ; . . . ." A3^-H32, pp. 5-62, ^* Les Secretz De V Apocalypse outierts et mis an iour.^^ 1\^ [p. 63], *^ Approba- tion" dated 20 June 1609. H42 [p. 64], ** Extraict du Priuilege du Roy,' dated 27 March 1610.

Signatures. A to H in fours = 32 leaves.

Paging. 64 numbered (except on first two and last two) = 64 pages.

From the title-page it would appear that a previous edition had been published at Rouen. The work is written to confute Napier's inter- pretation of the Apocalypse, and commences thus : —

Depvis quatre ou cinq ans, a este veu vn liure intitule, L'ouuerture de V Apocalypse^ mis en lumiere par Napeyr Escossois, duquel n'ay voulu pub- lier les erreurs, aduerty que ses partisans mesme le desauouet, come plein

de mensonges & impostures Croy certainement qu'en son oeuure

Sathan a voulu iouer sa reste ; Et voyant son temps si pres, nous enuoyer par ce docteur ses harmonyes Pythonissiennes, cauteleusement douces, & a la verite pleines d'attrait, pour nous pyper.

Libraries. Un. Ed.;

Le

Appendix. 151

A Rouen, I Par Lavrens Mavrry, rue neuve|S. Lo, a Timprim- erie du Louvre. | M. DC. LXV. | Avec Permission. |

12°.

Libraries. Brit. Mus. Lon.;

Aureum | Johannis Woltheri Peinensis Saxonis : | Das ist : | Gvlden Arch, Da- 1 rinn der wahre Verstand vnd Einhalt der | wichtigen Geheimnussen, Worter vnd Zahlen, in der | Offenbah- rung Johannis, vnd im Propheten Daniel, reichlich | vnd iiber- fliissig gefunden wird, Wie dann auch eine bewerthe Prob aller | Propositionen, vnd auszfuhrliche Wiederlegung, der vermeynten lang- 1 gewUnscheten Auszlegung iiber diese Offenbarung Johan- nis, desz Treffli- | chen Schottlandischen Theologi, Herrn Johan- nis Napeiri, durch | die Historien vnd Geschichten der zeit er- klaret | vnd angezeigt. Mehr wird auch darinn vor Augen gelegt I vnd dargestellt, wie iibcl vnd boszlich M. Paulus Na- | gelius mit dem Propheten Daniel vnd der Oflfenbahrung Jo- 1 hannis vmbgehe, vnd was von seiner, vnd der andern Newen | Rosencreutzbriider Astronomia gratiae oder Apocaly- 1 ptica zu halten sey. | Letzlich werden auch erortert H. Napeiri, Wolffgan-] gi Mayers, Leons de Dromna, vnd anderer Calvinisten | grobe Jrrthumbe von der Rechtfertigung eines armen | Sunders, auch anderen Glaubens Articuln | Nebenst auch einem kurtzen Dis- curs von den Kirchen- | Ceremonien, &c. |

Psal. 94. 1 Recht musz doch recht bleiben, vnd dem werden alle fromme Hertzen zufallen. |

Gedruckt zu Rostock, durch Mauritz Sachsen, In vorlegung | Johan Hallervordes Buchhandlers daselbst. 1623.I

[Printed in black and red.]

T 4

152 Appendix.

4°. Size 6 X 4 inches. Black letter. Ai^, Title-page. A 1 2, blank. A2I- B3I, 1 1 pages, Dedication to Kurfiirst Georg Wilhelm Markgraf von Brandenburg,

?>\g!\QA " Dattim Liechtetihagen den 15 Octobris des id^i.Jahres E, Churf.

Durchl. Gehorsamer Vnterthan yohamtes Woltherus Pfarrherr daselbsty On

B32 is the title of the German translation of 'A plaine discovery,' printed at Frankfurt in 1615. B4i-Nn32, pp. 1-272, The work itself [see note]. On Nn4i is

printed '^ Gedruckt zu Rostock \durch Moritz Sacksen,\Im Jahr CAristi \ 162^. \ " Nn42, blank.

Signatures. A to Z and Aa to Nn in fours =144 leaves. Paging. 14 + 272 numbered + 2 = 288 pages.

Johann Wolther, the Pastor of Liechtenhagen, was a zealous Lutheran, and adherent of the Augsburg Confession. In his work he reprints in full the 36 propositions of Napier's First treatise as given in the Frank- furt edition of 1615, pp. 1-122, omitting the * Beschluz,' p. 123. To each proposition is appended a refutation of the same.' These refuta- tions, being much longer than the propositions, form the bulk of the book.

Libraries. Un. Breslau :

Kunstliche Rechenstablein | zu vortheilhafftiger vnd leichter ma- 1 nifaltigung, Theilungwie nicht | weniger | Auszziehung der gevierdten vnd Cubi- 1 schen Wurtzeln, alien Rechenmeistern, In- I genieuren, Bawmeistern, vnd Land- 1 messern, vber die masz dienlich. | Erstlich 1617. In lateinischer Sprach durch Herrn Johan Nepern, Freyherrn in Schottlandt, | beschrieben, nacher ausz anleytung, desz hochgelehrten | weitberiihmbten Herrn v. Bayrn durch Frantz Keszlern zu Werck gericht. In Kurtz ver- fast, vnd zum Truck gefertigt. |

Gedruckt zu Straszburg bey Niclaus | Myriot, In verlegung Jacob von der Heyden Chal- 1 cographum | Anno MDCXVIII. | 4°. Size inches. Black letter.

Libraries. Stadt Frankfurt; K. Hof u. Staats Miinchen;

hergegen

Appendix. 153

Gedruckt zum Berlin im Grawen Kloster, durch George Run- gen, I Im Jahre Christi 1623. |

4°. Size 6^ X 4j inches. Ki^, Title. A i^, blank. A2i-A3\ 3 pages,

*' Vorrede an den guthertzigen Leser^ A3^-C4^, 18 pages, " Von der Stdhelrech-

WM«^, " containing Cap. I. " Vonbeschreibungvnd gebrauchder Stdblichen ins gemein.^'' ; Cap. II. •* Wie das Multipliciren mit Mil ff der Stdbichen verrichtet werdey ; Cap III. ** Wie das Dividiren anzustellen sey.^' ; Cap IV. ** Von erfindimg einer jeden Zahlen quadrat WurtzeV^ ; Cap. V. ** Wie man mit hiilffe des Bldtichen Pro Cubica, vnd der Stdbichen einer jedern Zahl radicem cubicam erfinden soiled C4^, ^^ Der Leser

wissey wo er der miihe die Stdbichen auffzutragen, wil vberhaben sein : das solche zier- lich in einem subtilen Kdstichen, aller notturff ncuh zugerichtet zubekommen sein. Vnd zufinden bey Martin Gut hen, Buchhdndlern zu Colin an der Spree^^

Signatures. A to C in fours =12 leaves.

Paging. 24 pages not numbered.

Facing A32 on a folding sheet is a diagram of the rods.

Libraries. Brit. Mus. Lon. ; Kon. Berlin ; Stadt. Breslau ; Nat. Paris ;

Another Edition. Gedruckt im Jahr Christi, | Anno 1630. | Libraries. Stadt. Zurich :

Manvale | Arithmeticae & Geometriae Practicae : | In het welcke I Beneffens de Stock-rekeninghe ofte | Rhabdologia J. Napperi cortelick en duydelic t' ge- 1 ne den Landmeters en Ingenieurs, nopende 't Land- 1 meten en Sterckten-bouwen nootwendich is,

V wort

154 Appendix.

wort I geleert ende exemplaerlick aenghewesen. | Op een nieu verrijckt met een nieuwe inventie on alle ronde va- 1 ten hare wannigheden af te pegelen. | Door | Adrianum Metium. Med. D. & Ma- I thes. Profess, ordinar. binnen Franeker. |

Tot Amsterdam, | By Henderick Laurentsz, Boeckvercooper op 't I Water, int Schryfboeck, Anno 1634. |

8°. Size inches.

Paging. 16 + 246 numbered + 8.

Another Edition.

Gedruckt by Ulderick Balck, Ordi- 1 naris Landschaps ende Academise Boecke- 1 Drucker. Anno 1646. |

Paging. 8 + 377 numbered + 1 1 .

These two editions are catalogued by D. Bierens de Hann in his papers entitled * Bouwstoffen voor de Geschiedenis der Wis- en Natuur- kundige Wetenschappen in de Nederlanden/ communicated to the Amsterdam Academy — Verslag. xii., 1878 (Natuurk.), p. 19.

The Art of | Numbring | By | Speaking-Rods :| Vulgarly termed I Nepeir's Bones. | By which | The most difficult Parts of | Arith- metick, | As Multiplication, Division, and Ex- 1 tracting of Roots both Square | and Cube, | Are performed with incredible Cele- 1 rity and Exactness (without any | charge to the Memory) by Addi- I tion and Substraction only. || Published by W. L. ||

London ; | Printed for G. Sawbridge, and are to be sold | at his House on Clerkenwell-Green, 1667. |

12°. Size 4fx2| inches. A i, blank? K2>, Title-page. A22, blank.

A3^-A6S 7 pages, " The Argument To The Reader:' MP; blank. After A62

on a folding sheet is a diagram of the rods. Bi^-E72, pp. 1-86, The Work.

E8\ ''Errata:' ES^-Eg^, 3 pages, Advertisements. E10I-E122, 6 pages,

blank?

Signatures. A in six + B to E in twelves=54 leaves.

Paging. 1 2 + 86 numbered + 10 = 108 pages.

The author was William Leyboum, and the work contains a short

description

I

Appendix. i^^

description of the rods, with examples of their use in multiplication, division, and the extraction of square and cube roots. Libraries, Un. Ed.; Brit. Mus. Lon. (2); Lambeth Pal. Lon.;

Another Edition.

London, printed by T. B. for H. Sawbridge, at the | Bible on Ludgate-Hill. 1685. |

Libraries. Un. Ab. ; Brit. Mus. Lon. ;

Nepper's Rechenstabchen, als Hiilfsmittel bei d. Multiplica- tion u. Division d. Zahlen- u. Decimalbriiche ; hrsgg. v. F. A. Netto. Mit 100 Rechenstabchen, Dresd. 181 5. Arnold. 18^.

This entry is copied from C. G. Kayser's Vollstandiges Biicher-Lexi- con (1750-183 2), pubHshed at Leipsic in 1835. The work apparently treats of 'Napier's Bones.'

8°. Size 6|x3i inches. ai^-H^, 20 pages, Preliminary matter. ai*-p2*,

pp. 1-116, ''Des Triangles Rectilignes:' A1I-Y4I, pp. I-I93 [l75] "-^^

Triangles Spheriqves. " ¥4^, woodcut.

In the first part, on a folio sheet facing p. 68, is a table of * Racines de 10' and of their logarithms.

Paging. 20+116 numbered + (193-18 for error =) 175 numbered + 1 = 312 pages.

Signatur:s. a in 4, e in 2 & i in 4 + a to o in 4 & p in 2 + A to Y in 4= 156 leaves.

^tJ 2 Errors

156 Appendix.

Errors in Paging. In first part none of consequence. In second part 168 numbered 186, and so to the end, thus making an error in excess of 18.

Permission to print the work was given on 5th April 1635. The Dedication is signed by the Author ' Iacobvs Hvmivs, Theagrius Scotus.' On the last page (p. 116) of the first part will be found the passage relating to Napier's burial-place, &c., part of which is quoted at p. 426 of the Memoirs. The two celebrated propositions by Napier are Nos. 117 & 120 of the second part.

Libraries, Adv. Ed. ; Un. Ed. ; Roy. Soc. Lon. ;

Tvbingae Apvd Haere | des Vlrici Morhardi. Anno | M.D. LIIII. I

[Printed in black and red.]

4°. Size 8 X 5f inches.

In describing the formation of the Logarithmic Table, in section 59 of the Constructio, Napier says that Reinhold's common table of sines (or any other more exact) will supply the values for filling in the natural sines in columns 2 and 6, and the table of sines* in this work (" Canoti Sinwm Vel Semis sivm Rectarvm In Circvlo SvbtensarvmJ\ fol 114), was probably the one he made use of.

Libraries. Trin. Col. Dub. ;

DN.

Appendix. 157

Typisq. exscriptam | Coloniae sumtibus Martini Guthij, | Anno CID IOC XVIII. I

[Note. Colonia=Kolna.d. Spree=BerHn.]

8°. Size 4| X 3 inches. Ai^, Title. Ai^-As^, 5 pages, Dedication to '' Illustri

et generoso domino^ Dn. Abrahamo lib. Baroni et Burggravio de Dohna "

signed **?« Valle nostra loachimica XVI. Kal. yun. anni seculi hujus XVII. T. Illustr. Generos. humiliml addictus Cliens Benjamin Ursinus." • A4*-C7', 40 pages, " Trigonometric Logarithmicce J. Neperi, dr^f." C8, 2 pages, blank. Aai\ The title, " Tabula Propor'\tionalis\Seqtienti\Canoni\Logarithmo-\rum Imer-\ viens. \ " Aai2-Aa5^ 8 pages, The Table. AaS^-Aay^, 5 pages, " Usus prcecedentis tabula.'' AaS, 2 pages, blank. Bbi\ The title, "y. Neperi \ Baronis Mer-\ chistonii, Sco- \ ti, dfc. \ Mirijicus \ Canon Logarith- \ morum. \ " Bbi'-Gg6^, 90

pages. The Canon to two places less than that of 1614. Gg6^-Hh2^, 9 pages,

'''■Lectori Benevolo,''' [errata.] The work should have ended on Hhi^, but through an error in printing, the two pages, Hhi^ and Hh2^, have been left blank.

Signatures. A to C and Aa to Gg in eights + Hh in two = 82 leaves.

Paging. 48 +16 + 91+9=1 64 pages not numbered.

Napier's Canon of 16 14 is here reprinted, but is shortened two places. Libraries. Brit. Mus. Lon.; Stadt. Breslau;

Another Edition. Coloniae, Martinus Guthius, 161 9. Libraries. Bodl. Oxf. ; Un. Camb. ; Nat. Paris ;

The First Edition

is stated to have been published in 161 7, which is no doubt correct, as the Dedication is dated 17th May 161 7.

V 3 Coloniae

158 Appendix.

Coloniae | Sumptib. M. Guttij, tipys | G. Rungij descripta. | CID IDC XXV.|

[The above is engraved on a half-open door, forming the centre of a title-page elaborately engraved by Petrus RoUos.]

4°. Size 7ix6| inches. ):( i^, Title. ):( i^, blank. ):( 2^- ):( 4^, 6 pages, Dedication to Dn. Georgio Wilhelmo Marchioni Brandenburgico, dated 1624. Ai^- L^'^, pp. 1-272. Trigonometria, in three books. Liber I., *'Z>^ TrianguliSy eorumq. affectionibiis.^* ; Sectio Prior, ^^ De Triangulis Planish ; Sectio Posterior, "Z?^ Triangulis Spharicis." . Liber II., '^ De Constructione Canonis Triangulorum ; ejusq. usu in genere." 'y Sectio I., *' De Constructione Canonis Sinutim.^^ -, Sectio II., ** De Constructione Tabulce Logarithmorum." ; Sectio III., "Z>^ usu Canonis Loga- rithmorum in genere.^' . Liber III., **Z>^ Usu Canonis Logarithmorum in utraq. Trigonometridy 'y Sectio I., ^* De Mensuratione Triangulorum Planorum sive Recti- lineorum." ; Sectio II., *^ De Trigononietrid SphcBricorum."

Signatures. ):( and A to Z and Aa to LI in fours = 140 leaves. Paging. 8 + 272 numbered = 280 pages.

Keppler. Harmonic. Lib. iv. cap. vii. p. 168. — [followed by extract of 8 lines.]

Coloniae, | Typis Georgij Rungij, impensis & sumtibus Martini Guttij I Bibliopolai, Anno M. DC. XXIV. |

Ai\ Title. A 1 2, blank. A2I-LII22, The Table occupying 450 pages.

Lll3\ *^ Emendanda in Canone," 35 lines. On LII32 is printed *^ Berolini, \

Excudebat Georgius Rungius TypographuSy \ impensis dr" sumtibus Martini Guttij \ Bibliopolce Coloniensis. \ Anno d'd Ico XXIV. | ". LII4, blank.

Signatures. A to Z and Aa to Zz and Aaa to Lll in fours = 228 leaves.

Paging. 2 + 450 + 4 = 456 pages not numbered.

Colonia, the place of publication, is Koln a. d. Spree or Berlin.

The second and third books of the Trigonometria deal with the sub- jects treated of in Napier's Descriptio and Constructio, these works being largely made use of by Ursinus, who speaks of Napier as a Mathematician without equal (see p. 131, 1. 5). The references in the text are to the Lyons edition of 1620 (see p. 178).

The Magnus Canon contains the logarithms of sines for every 10" in the quadrant. They are arranged in a similar way, and are of the same kind as those in Napier's Canon of 16 14,* but are carried one place

further,

Appendix.

159

Grad. 30.

M.S. Stmts. D. Logarith. D. Different. D. Logarith. D. Sinus. D.

O o 10 20

I o 10 20

2 o 10 20

3 ° 10

20

4 ° 10

20

5 ° 10

20

60

50000000 04199 08397

12595 16794 20992

25190 29387 33585

37783 41980 46178

50375 54572 58769

62966 67163 71359

75556 79752

83949 88145 92341 96537

50100733 04928 09124

13320

17515 21710

25905 30100 34295

38490 42685 46879

51074

99

69314718 06321 69297925

95

89530 81136

72743

64351 55960

47570

39181 30792 22405

14019 05633 69197249

88866 80483 72101

63720 55340 4696

38583 30206 21830

13455 05081 69096708

88336 79964 71593

63224

54855 46487

38121

29755 21390

13026

54930614 19417 08222

54897027 85833 74640

63447 52255 41064

29873 18682 07493

54796304 85115 73928

62741

51554 40368

29182 17997 06813

54695630 84447 73265

62083 50902 39722

28542

17363 06184

54595007 83829 72652

61476 50301 39126

27952

87

I 4384104 86904 89703

92503 95303 98103

14400904

03705 06506

09308 12110 14912

17715 20518 23321

26125 28929 31733

34538 37343 40148

42953 45759 48565

51372

54179 56986

59794 62601 65409

68217 71026 73835

76645 79454 82264

85074

02

03

04

05

06

07

09

86602540

00116

86597692

95267 92842 90417

87992 85567 83141

80716 78290 75863

73437 71010 68583

66156

63729 61302

58874 56446 54018

51590 49162 46733

44304 41875 39446

37016 34587 32157

29727 27296 24866

22435 20004

17573

15142

24

28

29

60

50 4o_

30 20 10

059 50 40_

30 20 10

058 50 40_

30 20 10

057 50 40^

30 20 10

056 50 40^

30 20 10

50 40^

30 20 10

°54

Grad. 59.

S.M.

V 4

i6o Appendix.

further, radius being made 100,000,000. The entire Canon was recom- puted by Ursinus, and full details of its construction are given in Book II., sect. 2, of the Trigonometria. The methods employed are the same as those laid down in the Constructio with the modifications in regard to the preliminary tables proposed by Napier in sect. 60. A specimen page of the Table is given on the preceding page, and refer- ence may also be made to my notes, pp. 94, 95.

Libraries. Un. Ed. ; Bodl. Oxf. ; Brit. Mus. Lon. ; Stadt. Breslau ; Nat Paris;

Johann Carl Schulze | wirklichen Mitgliedes der Konigl. Preussischen Academic der | Wissenschaften | Neue Und Erweit- erte | Sammlung | Logarithmischer, | Trigonometrischer | und anderer | Zum Gebrauch Der Mathematik | Unentbehrlicher | Tafeln. || II. Band. ||

Berlin, 1778. | Bey August Mylius, Buchhandler | In Der Briiderstrasse. |

Size 8| X 6 inches.

In this work the logarithms of the Magnus Canon of Ursinus are reprinted to every 10 seconds in the case of the first four and last four degrees, being the same as in the original. The logarithms from 4° to 86° are given for every minute only. Ursinus' logarithms occupy half the lower portion of pp. 2-261 in Volume II., the title of the whole con- tents of these pages being : —

•* Tafel I der \ Sinus, Tangenten, \ Secanten \ und\ deren zustimmenden briggischen und hyperboli-\schen Logarithmen \fur die vier ersten und vier letzten Grade von 10 zu\\o Secunden ; \fiir den iihrigen Theil des Quadranien aber von Minute zu \ Minute^ nebet dem dten Theile der Differenzen \ berechnet. \ "

nee

Appendix. i6i

Marpurgi,|Excusa Typis Casparis Chemlini.|cIo loc xxiv.|

4". Size 8x6^ inches. Ai^, Title. Ai^, blank. Folding sheet with

*• * * -^Ad Postul 2. Exemplvrn SecHonis, ' ' ' *". A2^, p. 3, Dedication by

Kepler to Philip Landgrave of Hesse. A22-F3I, pp. 4-45, " Demonstratio Strvc-

tvrcB Logarithmorvmy in 30 propositions. Y-^-Q^, pp. 46-55, *^Methodvs Com-

pendiosissima constniendi Chiliada Logaritkmorum.^^

On G42 is the title ** Chilias \ Logarithmcrum \ Joh, Kepleri, Mathem. \ Casarei. \ " H 1^-022, 52 pages occupied by the table, and at the foot of the last page ** Errata," 10 lines.

Signatures. A to N in fours + O in two = 54 leaves.

Paging. 55 numbered+ 1 +52= 108 pages, also folding sheet.

Signature O is distinctly in two, the work ending with p. 108, but the Supplementum assumes it to end with p. 112, which it would have done had sig. O been in four.

Marpvrgi, | Ex officina Typographica Casparis Chemlini. | cIdIocXXV.I

Pi\ p.[ii3], Title. Pi2, blank. P2I-P32, pp. 113 [115]-! 16 [118], 4 pages,

*^ Joannis Kepleri Suppletnentum Chiliadis Logarithmoruniy Continens Prcecepta De Eorum Usu. Lectori S." ¥4^, P-[il9] " Cortectio Figuraruni post punctum in

Logarithmis.'''* P42, p. [120] '^ Prceterea in textu Demonstratimum jam im-

presso, notaviista, nondum d. Typographo atiimadversa.'' 8 lines of corrections. Qii-Dd42, pp. 121-216. The work in 9 chapters. The pages are all headed "Joannis Kepleri Chiliadis Complement," not Supplement.

Signatures. P to Z and Aa to Dd in fours = 52 leaves.

Paging. P. [113] to p. 216=104 pages.

Errors in Paging. Pages 115 to 1 18 containing the Preface are numbered in error 113 to 116.

The first part of the work contains Kepler's demonstration of the structure of logarithms, which is in form geometrical, some of the Ger-

X nian

l62

Appendix.

A R c u s Circuit cum differentiis.

3. 55

29. 0.45

3. 56

29. 4.41

3. 56

29. 8. 37

3. 56

29. 12. 33

-3. 57

29. 16. 30

3. 56

29. 20. 26

3. 57

29. 24. 23

3- 57

29. 28. 20

3- 57

29. 32. 17

3. 58 29. 36. 15 3. 57

29. 40. 12

3- 57

29.44. 9

3. 58

29.48. 7

3- 57

29. 52. 4

-3. 58

29. 56.

3.

30. O.

3.

30.

2 58 O 58 3.58 3. 59 30. 7- 57

3. 58

30. II. 55

3. 59 30. 15. 54

3. 59

Sinus

feu Numeri

abfoluti.

48500. 00 48600. 00 48700. 00 48800. 00 48900. 00 49000. 00 49100. 00

49200. 00 49300. 00 49400. 00

49500. 00

49600. 00 49700. 00 49800. 00 49900. 00 50000. 00 50100. CO 50200. 00 50300. 00 50400. 00

Partes vicefima quarta.

	38.24 . 39. 50
	.41. 17 . 42. 43
	.44. 10 . 45. 36
	47. 2 48. 29
	49.55 51. 22
	52.48 54. 14
	55.41 57. 7
12	58.34 0. 0
12. 12.	1.26 2. 53
12. 12.	4. 19 5.46

LOGARITHMI

cum differentiis.

206. 40 72360.64

205. 97 72154.67

205. 55 71949.12

205. 13

71743-99 204. 71

71539.28 204. 29

71334.99 203. 87

71131.12 —

203. 46 70927.66

203. 05 70724.6 1 -F

202. 63 70521.98

202. 23

70319.75H-

201. 81 70117.94—

201. 41 69916.53—

201. 01 69715.52

200. 60 69514.92

200. 20 69314.72

199, 80 69114.92

199. 40 68915.52 —

199. 01 68716.51-+

198. 61 68517.90-h

198. 21

L 2

Partes fexage- naria.

29. 6

29. 10

29- 13 29. 17 29. 20 29. 24 29. 28 29. 31

29. 35 29. 38 29. 42

29. 46 29.49

29.53 29.56

30. o 30. 4

30- 7 30. II 30. 14

Arcu s

Appendix. 163

man mathematicians, as he mentions in his Preface, not being satisfied with Napier's demonstration based on Arithmetical and Geometrical motion. The two parts together with the Table are reprinted in *Scriptores Logarithmici,' vol. I. p. i. At the beginning of the same volume is reprinted the Introduction to Hutton's Mathematical Tables, on p. liii of which will be found a "brief translation of both parts, omitting only the demonstrations of the propositions, and some rather long illustrations of them."

The logarithms in the Table are of the same kind as Napier's, but they are not affected by the mistake in the computation of the Canon of 1 6 14.

The Tables of Kepler and Napier are differently arranged, and the numbers for which the logarithms are given are also different. In Napier's Canon the numbers in column "Sinus" are the values of sines of equidifferent arcs, while in this table the numbers or sines are equidifferent. For specimen page of the Table see preceding page. The arrangement is as follows : —

Column 2 contains looo equidifferent numbers, 10,000, 20,000, 30,000, . . . 9,980,000, 9,990,000, 10,000,000. It also has at the be- ginning the 36 numbers i, 2, 3, to 9; 10, 20, 30 to 90; 100, 200 to 900; and 1000, 2000 to 9000.

Column 4 contains the logarithms of the numbers in column 2, with interscript differences.

The 2nd and 4th are the only columns containing entries for the first 36 numbers.

It will be observed that a point m.arks off the last two figures of the values in these two columns, but if it be left out of account the numbers and logarithms agree with those of the Canon of 16 14, in being referred to a radius of 10,000,000. So that the values really represented are the ratios of the numbers there given to 10,000,000.

Taking as an example the first entry in the specimen page, the num- ber in column 2 which is 4,850,000 represents the ratio 4,850,000 to 10,000,000 or a T^M^^^th = a T^f ^th part of radius. Similarly column I gives the arc, in degrees, minutes, and seconds, corresponding to a sine equal to the i^Mth part of the radius, with interscript differences ;

Column 3 gives in hours, minutes, and seconds the x^fjth part of a day of 24 hours ; and finally

X 2 Column

164 Appendix.

Column 5 gives in minutes and seconds the y^f fth part of a degree of 60 minutes. Libraries. Sig. Ed.; Un. Gl.; Hunt. Mus. Gl.; Bodl. Oxf.; Trin. Col. Dub.;

Tabvlae Rudolphinae. . . . loannes Keplerus. . . .

Ulmae. Jonae Saurii. Anno M.DC.XXVII. Folio. Size 13| x 9 inches.

The logarithms used in this work are those of Napier. Libraries. Adv. Edin. ; etc.

LOGARITHMORVM | ChILIAS PrIMA. |

Quam autor typis excudendam curauit, non eo con- | cilio, vt publici iuris fieret ; sed partim, vt quorun- 1 dam suorum neces- sariorum desiderio priuatim satis- | faceret partim, vt eius adiu- mento, non solum Chilia- 1 das aliquot insequentes ; sed etiam integrum Loga- | rithmorum Canonem, omnium Triangulorum cal- 1 culo inseruientem commodius absolueret. Habet e- 1 nim Canonem Sinuum, k seipso, ante Decennium, per | aequationes Algebraicas, & differentias, ipsis Sinu- 1 bus proportionales, pro singulis Gradibus & graduu | centesimis, a primis fundamentis accurate extructu : | quem vna cum Logarithmis adjvnctis, vol- ente Deo, | in lucem sedaturum sperat, quam primum commode | licuerit. |

Quod autem hi Logarithmi, diversi sint ab ijs, | quos Clarissi- mus inuentor, memoriae semper colendae, | in suo edidit Canone Mirifico ; sperandum, eius libru | posthumum, abunde nobis pro- pediem satisfactu- 1 rum. Qui autori (cum eum domi suae, Edin- burgi, I bis inuiseret, %l apud eum humanissime exceptus, | per aliquot septimanas libentissime mansisset ; eique | horum partem praecipuam quam turn absoluerat | ostendisset) suadere non des-

titit,

Appendix. 165

titit, vt hunc in | se laborem susciperet Cui ille non | inuitus morem gessit. |

In teniii ; sed non tenuis^ structusve laborve.

8°. 16 pages.

The above short Preface occupies the first page of a small tract of sixteen pages, the remaining fifteen containing the natural numbers from I to 1000 with their logarithms, to base lo, to 14 places. The tract bears no author's name or place or date of publication, but the evidence which assigns it to Briggs, and fixes the place and date of its publication as, London, 161 7, seems conclusive. The Table of Logarithms is the first published to a base different from that employed by Napier.

It is unnecessary here to refer to subsequent works on Logarithms of a different kind from those originally published by Napier.

Libraries, iBrit. Mus. Lon. :

Note. — In the foregoing Catalogue the only collections of Napier's works referred to are in public libraries. The largest single collection, however, is that in possession of Lord Napier and Ettrick. Besides the editions more commonly met with, it embraces several not found in any of the public libraries of this country, as well as a copy of the rare ^Ephemeris Motuum CoelesHum ad annum 1620,' which contains Kepler's letter of dedication to Napier, dated 27th July 161 9.

SUMMARY OF CATALOGUE.

A Plaine Discovery. In English.

Edinburgh, by Robert Waldegrave, 1593

Variety, with part of Sig. B reset London, for John Norton, 1594 Edinburgh, by Andrew Hart, 161 1 London, for John Norton, 161 1 Edinburgh, for Andro Wilson, 1645

PAGE 109 IIO

III

112

In Dutch.

Translation by " Michiel Panneel, Dienaer des Godelijcken mwrts tot Middelborch."

Middelburgh, by Symon Moulert, 1600 . . . .115

Middelburch, voor Adriaen vanden Vivre, 1607. Translation re- vised, with additions, by G. Panneel . . . . 116

In French.

Translation by Georges Thomson.

La Rochelle, par Jean Brenouzet, 1602 . . . .118

The same, with substituted title-page. La Rochelle, pour

Timothee Jovan, 1602 . . . . . .119

La Rochelle, par les Rentiers de H. Haultin, 1603 . . .120

La Rochelle, par Noel de la Croix, 1605 .... 121

La Rochelle, par Noel de la Croix, 1607. The Second Treatise ends

on p. 406 ....... 122

Variety, with difference in title-page of the Quatre Harmonies . 123 La Rochelle, par Noel de la Croix, 1607. The Second Treatise ends

on p. 392 . . . . . . . 123

In

Summary of Catalogue.

In German.

Translation of the First Treatise only by Leo de Dromna.

Gera, durch Martinum Spiessen, 1611

The same, but with date 1612 • . . !

Translation of the First and Second Treatises by Wolfpgano Mayer.

Franckfort am Mayn, 1615 .....

Franckfurt am Mayn, 1627 .....

De Arte Logistica. In Latin. Edinburgi, 1839. Club copies and large paper copies

Rabdologiae. In Latin. Edinburgi, Andreas Hart, 161 7 . . .

The same, with error in title-page corrected Lugduni, Petri Rammasenii, 1626 ....

The same, with substituted title-page. Lugd. Batavorum Petri Ramasenii, 1628 .....

In Italian. Translation by "II Cavalier Marco Locatello."

Verona, Angelo Tamo, 1623 .....

In Dutch.

Translation by Adrian Vlack.

Goude, by Pieter Rammaseyn, 1626 ....

167

125 126

126 128

129

131 131 132

132

133

134

Works on Logarithms. In Latin. Descriptio. Edinburgi, Andreae Hart, 1614. With Admonitio

The same : varieties without Admonitio . Descriptio reprinted in Scriptores Logarithmici, vol. vi. London

R. Wilks, 1807

Constructio. Edinburgi, Andreas Hart, 1619

Descriptio and Constructio. New title-page for two works, Edinburgi, Andreas Hart, 1619 Descriptio and Constructio. Lugduni, Barth Vincentium, 1620 Variety, with title-page of Descriptio dated 1619 . The same, but sig. A of Descriptio reprinted. Lugduni, Joan Anton. Huguetan & Marc. Ant. Ravaud, 1658

X 4

137 139

139 140

140 141

M3

144 In

i68 * Summary of Appendix.

In English.

The Descriptio translated by Edward Wright.

London, by Nicholas Okes, 1616 .... 144

The same, with substituted title-page. London, for Simon

Waterson, 161 8 . . . . 146

Retranslated by Herschell Filipowski.

Edinburgh, by W. H. Lizars, 1857 . . . . . 147

SUMMARY OF APPENDIX.

Napier's Narration. London, for Giles Calvert, 1641 . . 148

The Bloody Almanack. London, for Anthony Vincent, 1643 . 149

The same, with additions ...... 149

A Bloody Almanack, 1647 ...... 149

Le Sommaire, par le Sieur de Perrieres Varin. Paris, Abraham le

Feure, 1610 . . . . . . .150

A previous edition, published at Rouen .... 150

Le Desabusement, par le Sieur F. de Courcelles. Rouen, Laurens

Maurry, 1665 ....... 151

Gulden Arch, by Johannes Woltherus. Rostock, Mauritz Sachsen,

1623 ....... 151

Kiinstliche Rechenstablein, by Frantz Keszlern. Straszburg, Nic-

laus Myriot, 1618 ...... 152

Rhabdologia Ncperiana, by Benjamin Ursinus. Berlin, George Run- gen, 1623 . . . . . . .152

Another edition. Anno 1630 ..... 153

Manuale Arithmeticae & Geometriae Practicae, by Adrianus Metius.

Amsterdam, Henderick Laurentsz, 1634 . . '153

Another edition. Ulderick Balck, 1646 . . . .154

The art of numbring by speaking-rods, by W. L. London, for

G. Sawb ridge, 1667 . . . . . .154

Another edition. London, for H. Sawbridge, 1685 • • ^55

Nepper's Rechenstabchen, by F. A. Netto. Dresden, Arnold, 1815 . 155

Traitd

Summary of Appendix.

169

Traits de la Trigonometrie, by Jacques Hume. Paris, Nicolas et Jean

de la Coste, 1636 .....

Primus Liber Tabularum Directionum, by Erasmus Rheinholdus,

Tubingse, Haere des Ulrici Morhardi, 1554 . Cursus Mathematici Practici Volumen Primum, by Benjamin Ur

sinus. First edition, 1617 .... The same. Coloniae, Martini Guthii, 1618 The same. Coloniae, Martinus Guthius, 1619 Trigonometria and Magnus Canon by Benjamin Ursinus. Coloniae,

Georgii Rungii, 1625 and 1624 Neue und erweiterte Sammlung Logarithmischer .... Tafeln, by

Johann Carl Schulze. Berlin, August Mylius, 1778 Chilias Logarithmorum and Supplementum, by Joannes Keplerus

Marpurgi, Casparis Chemlini, 1624 and 1625 Tabulae Rudolphinae, by Joannes Keplerus. Ulmae, Jonae Saurii,

1627 ........

Logarithmorum Chilias Prima, by [Henry Briggs]. [London, 1617.]

»55 156

157 156

157

160

164 164

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PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET

UNIVERSITY OF TORONTO LIBRARY

Qk Napier, John

33 The construction of the

N35S wonderfiil canon of logarithms

1889

Physical k Applied ScL