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THE LIBRARY

UNIVERSmr OF GUELPH

Date due

DE ARTE LOGISTICA.

nti'«'T^fI,.H>I.lp!.-

DE AETE LOGISTICA

JOANNIS NAPEEI

MERCHISTONII BARONIS

LIBRI QUI SUPERSUNT.

IMPRESSUM EDINBURGI
M.DCCC.XXXIX.

THE LIBRARY
UNlVtRSlTY OF 6UELPM

At a Meeting of the Committee of the Bannatyne Club, held in the
House of the President, on Monday the 28th of January, 1839> —

*' Resolved, That One Hundred and One Copies of a Volume,
entitled Joannis Neperi A Merchistoun Baronis de Arte Logistica
LiBRi Tres, be subscribed for in the name of the Club, to MARK
NAPIER, EsQ., Advocate ; and that Club paper be furnished for that
number."

Extracted from the Minutes ofthe Club.

DAVID LAING, Secretary.

THE BANNATYNE CLUB.

SEPTEMBER, M.DCCC.XXXIX.

THOMAS THOMSON, ESQ.

PRESIDENT.

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100 THE VEN. ARCHDEACON WRANGHAM.

n
TO THE RIGHT HONOURABLE

FMNCIS LORD NAPIER OF MERCHISTON,

ETC. ETC. ETC.

My dear Lord Napier, *

This Memorial of your great Ancestor, which it has
been my ambition to present in a form worthy of the genius it
records, I dedicate to you, in remembrance of the exemplary
liberaKty with which the Archives of your noble family have
been always open to illustrate the History and the Letters
of Scotland.

Yours affectionately,

MARK NAPIER.

Edinburgh, November \, 1839.

INTEODUCTION.

a

INTRODUCTION.

In the Memoirs of Napier of Merchiston, published in 1834,
some account was given of two manuscript treatises — one of
Arithmetic, and the other of Algebra — composed in Latin by
that celebrated mathematician, and which had remained in-
edited in the charter-chest of his family, and indeed unknown
to the world, until the Memoirs were pubHshed. Upon that
occasion, Httle more could be afforded than a very imperfect
review of their contents. The idea subsequently occurred,
that it might gratify the lovers of science if these mathematical
studies of (to adopt the expressions of the historian Hume)
" The celebrated Inventor of Logarithms, the person to whom
the title of Great Man is more justly due than to any other
whom his country ever produced," — ^were added as an appendix

!▼

INTRODUCTION.

to a new edition of his Life. I have been induced, however,
to publish the treatises in their present independent and more
becoming form, by the spirited interposition of the Bannatyne
and Maitland Clubs of Scotland ; whose unanimous patrpnage
of the work, — with their characteristic care for, and pride in,
the ancient letters of Scotland, — ^has alone enabled me to render
the volume so worthy a memorial of Scotish genius.

Napier's scientific manuscripts came into the possession,
not of his eldest son Archibald, the first Lord Napier of
Merchiston, but of a younger son Robert. From the conge-
niality of their pursuits, Robert became his father's literary
executor, and edited his posthumous work, entitled " Logarith-
morum Canonis Constructio," being his secret of the construc-
tion of that Canon of the Logarithms which he had published
three years before his death. Robert Napier, designed of Bo-
whopple, Culcreugh, and Drumquhannie, was the second son of
Napier'8 second marriage. The late Colonel Milliken Napier,
Robert'8 lineal male representative, was still in possession of
many private papers of the family at the close of last century.
Upon one occasion, when the Colonel was called from home on
the service of his country, these papers, along with a portrait
of the great Napier, and a Bible with his autograph, were
deposited for safety in a room of the house of Milliken, in

INTRODUCTION. V

Renfrewshire. During the owner's absence the house was
burned to the ground, and the precious reUcs perished.

As these manuscripts had not been arranged, nor properly
examined, the extent of the loss is unknown ; but it is more
than probable that among them were the originals from which
Napier's son Robert made the transcripts accidentally pre-
served, and now given to the pubhc. The manner in which
these transcripts had escaped the fate of the Culcreuch papers
appears from the following note, written in the MS. volume
itself, by Francis seventh Lord Napier, grandfather of the
present Peer to whom it now belongs : —

" John Napier of Merchiston, inventor of the Logarithms,
left his manuscripts to his son Robert, who appears to have
caused the following pages to have been written out fair from
his father's notes, for Mr Briggs, Professor of Geometry at
Oxford. They were given to Francis, the fifth Lord Napier,
by WilHam Napier of Culcreuch, Esq., heir-male of the above-
named Robert. Finding them, in a neglected state, amongst
my family papers, I have bound them together, in order to
preserve them entire.

" NAPIEK."

" Jth Marchy 1801."

^ INTRODUCTION.

The transcripts are entirely in the handwriting of Robert
Napier himself, as is aseertained beyond doubt by a eomparison
with 8ome of his letters in Lord Napier's charter-chest ; and
this title, written on the first leaf, is also in his handwriting :—

" The Baron of Merchiston his booke of Arithmeticke
and Algebra. For Mr Henrie Briggs, Professor of Geometrie
at Oxforde."

That they were taken directly from the author'8 own
papers, is proved by the note which the transcriber adds at the
conclusion of the fragment on Geometrical Logistic : —

" I could find no more of this Geometricall pairt amongst
all his fragments."

And also by another note which immediately follows the
abrupt termination of the Algebra : —

" There is no more of his Algebra orderlie sett doun."

These notes, of which fac-similes will be found in their
respective places in this volume, seem to have been addressed
to Henry Briggs ; and they not only prove that the transcripts
were made from Napier's private manuscripts, but that Napier
himself had so far written out the treatises in the form now
published, and that more remained to be digested and arranged.

INTRODUCTION. vii

The late Lord Napier, whose untimely death in China, a
few years ago, attracted the mournfiil interest of the pubUc,
intrusted to myself, before his departure on that mission, the
many curious and valuable historical rehcs of his family, with
fiill permission to pubhsh whatever might serve to illustrate
the history and letters of Scotland. This task I have now
accomphshed, with at least industrious zeal, in the Memoirs
of Merchiston, the History of the Lennox, the Memoirs of
Montrose, and, finally, the editing of this beautiful, though
unfinished treatise, De Arte Logistica.

The authenticity of the MS., as an unpubhshed work of the
Inventor of Logarithms, being thus unquestionable, the date
of its composition becomes an interesting enquiry, in reference
to the history of science in Scotland. As Napier died in the
year 1617, no later date can be assigned to thesa mathematical
studies than the commencement of the seventeenth century —
an epoch when the hght of science is only said to have dawned,
in our country, because of the pubhcation of his own invention
of Logarithms. From the internal evidence, however, there is
every reason to suppose, that, when so much of his Arithmetic
and Algebra was thus " orderhe sett doun" by Napier, the
system of Logarithms was not only unknown to the world,
but had not as yet been developed in the mind of the inventor

yiii INTRODUCTION.

himuielf. Consequently, a much earlier date must be assigned
to the compositions now published. They are, in all proba-
bility, the first-fruits of that immortal genius which eventually
drew from the mysterious depths of Numbers their pearl of
greatest price. Napier's Arithmetic and Algebra will be re-
garded with reverential interest by all who can, in any degree,
appreciate the analytical power, and the practical effects of the
Logarithms ; especially if, in those treatises, we can trace the
preparatory labours of this secluded Scotish Baron, ere he
80 unexpectedly displayed himself on a pinnacle of science to
which Kepler himself paid homage.

Napier published his great work, entitled " Mirifici Canonis
Logarithmorimi Descriptio," in the year 1614. By that time
his mind was so deeply engrossed with the subject, and he was
80 constantly. employed, — during the short interval between
the date of that publication and his death in 1617, — with his
plans for extricating science from the trammels of ordinary
calculation, that his Arithmetic and Algebra can hardly be
imagined to have been his latest compositions. But what
appears decisive upon the point is, that there cannot be dis-
covered throughout the whole of the work now printed an
allusion even to the principle of the Logarithms. A complete
exposition of the subject was not, indeed, to be expected in his

INTRODUCTION. ix

treatise of Arithmetic ; but neither is it to be supposed that
he would have written, for instance, the fifth chapter of his
second book of Arithmetic, " De MultipUcationis et Parti-
tionis Compendiis Miscellaneis," after having pubhshed the
Logarithms, without the shghtest allusion to that means of
substituting, for the real numbers, a set of artificial numbers
rendering the whole art of calculation compendious beyond the
most sanguine hopes of emancipated Astronomy.

It is even more improbabie that the treatises in question
were composed, or so far reduced to order, at some period in
the long interval which must have elapsed between the author's
conception of the Logarithms, and the construction of those
elaborate works by which the glory of the invention was
secured to himself, and the benefit bestowed upon mankind.
In his book of Geometrical Logistic, Napier seems to allude
to the Arithmetic of Surds as a discovery of his own, the full
value of which he promises afterwards to disclose. Had he
been conscious at this time of the far more important secret of
the Logarithms, — which, after his first conception of the idea,
was his daily labour to the end of his days, — there would, in
all probabiHty, have been some allusion to it. Indeed, his
great invention must have formed a most interesting chapter
of his work De Arte Logistica, had he Hved to complete it.

b

X INTRODUCTION.

Not only is there no mention of his Canon of the Loga-
rithms in these treatises, but when, to illustrate the relation
between the powers of numbers and their indices, there is
exhibited the following table, —

I. II. III. IIII. V. VI. VII.
1. 2. 4. 8. 16. 32. 64. 128,—

no allusion occurs to the fact, that this table also afFords an
illustration of the logarithmic principle. The upper series
are not only indices to the lower series, considered as powers,
but they are Logarithms to those numbers, being an arith-
metical adapted to a geometrical progression. The word
Logarithms was compounded by Napier before he published
his discovery, and this term itself embodies the principle
illustrated (though on a very limited scale) by the above
arrangement of figures. A^idfMt signifies numbers ; Xoyu^tdfjuoi^
the ratios of numbers, or rather the number of ratios, Xoym
u(t6fM(. Now, although Napier expounded his system of
Logarithms through his geometrical idea of fluxions (a de-
monstration and a term afterwards adopted by Newton), and
not through the doctrine of powers and exponents, yet he
could not have composed the term Logarithms without a
perfect knowledge of the connexion of his system with the
arithmetic of proportions.

INTRODUCTION. xi

It can scarcely be doubted, then, that the composition of
Napier's treatise of Numbers is prior in date even to his
conception of the Logarithms ; a conclusion that will seem very
natural to any one who has examined his works on the subject,
and who is at all capable of appreciating the command of the
science of Numbers which those works evince. Undoubtedly,
his great invention was not (as is alleged of the theory of
gravitation) the result of a sudden thought or suggestion, in
some happy moment of genius. It was laboriously ehcited
from the arcana of Numbers, — deUberately extracted, by a
sort of Csesarean operation, from the unripe womb of analytical
science. Napier himself alludes to this predetermination of
his genius, in a most interesting letter, the last he ever wrote.
It is prefixed as a dedication, to the Chancellor of Scotland,
of his Rhabdologia, or the art of computing by means of
figured rods, better known by the name of " Neper's bones."
In the present attempt to trace the chronological order of his
works, I may here take the hberty to translate this letter from
the original Latin, the language in which he composed all his
mathematical works : —

jji INTRODUCTION.

" To the most illustrious Alexander Seton, Earl of
Dunfermline, Lord of Fyvy and Urquhart, High
Chancellor of Scotland, &c.
" The difficulty and prolixity of calculation (most illustrious
Sir), a toil which is apt to deter most people from the study
of raathematics, I have, all my life, with what powers and little
genius I possess, laboured to eradicate. And, with that end in
view, I pubHshed of late years the Canon of Logarithms (for
a long period elaborated by me), which, rejecting the natural
numbers, and the more difficult operations performed by them,
substitutes in their place others aflTording the same results by
means of easy additions, subtractions, and extractions of roots.
Of which Logarithms, indeed, I have now found out another
species much superior to the former, and intend, if God shall
grant me longer life, and the possession of health, to make
known the method of constructing, as well as the manner of
using them. But the actual computation of this new Canon, on
account of the infirmity of my bodily health, I have left to some
who are well versant in such studies; and especially to that
most leamed man, Henry Briggs, public professor of geometry
in London, my most beloved friend. Meanwhile, however, for
the sake of those who may prefer to work with the natural
numbers as they stand, I have excogitated three other compen-
dious modes of calculation. The first is by means of numerating

INTRODUCTION. xiii

rods, and this I have called Rhabdologia. Another, by far the
most expeditious of all for multiplication, and which, on that
account, I have not inaptly termed the Promptuary of Multi-
PLiCATioN, is by means of Httle plates of metal disposed in a
box. And, lastly, a third method, namely, Local Arithmetic
performed upon a chess-board. But, to the pubHcation of this
Uttle work, concerning the mechanism and use of the rods, I
was specially impelled, not merely by the fact that they are so
approved of as to be already almost in common use, and even
carried to foreign countries ; but because it also reached my
ears that your kindness advised me so to do, lest they should
be pubhshed in the name of another, and I be compelled to
sing with Virgil,

' Hos ego versiculos feci,' &c.

This very friendly counsel from your Lordship ought to have
the greatest weight with me ; and most assuredly, but for that,
this little treatise of the use of the rods (to which the other two
compendious methods are added, by way of appendix), would
never have seen the Hght. If, therefore, any thanks be due from
the students of mathematics for these little books, they all belong
to you as your just right, my noble Lord ; to you, indeed,
they must spontaneously fly, not only as patron, but a second
parent : especially since I am assured that you have done these

xiv INTRODUCTION.

rods of mine such high honour as to have them framed, not of
common materials, but of silver. Accept, then, my Lord, in
good part, this small work, such as it is ; and, though it be not
worthy of so great a Maecenas, take it under your patronage
as a child of your own. And so I earnestly pray God to preserve
you long, to us and the state, to preside over justice and equity.
" Your Lordship's very much obliged

"JOHN NAPIER,
" Baron of Merchiston."

The date of the volume to which this letter is prefixed is
1617, and Napier died upon the 4th of April in that same year.
We learn from it the chronological order of the composition
of all his published works. In the first place, he had long and
laboriously wrought out — " a me longo tempore elaboratum" —
his Canon of the Logarithms. Then he excogitated — " exco-
gitavimus" — ^the mechanical system of calculating rods, for the
sake of those who might be distrustiul of his artificial system
of Logarithms. His Promptuary of Multiplication, which
immediately follows the Rhabdologia, he states to be the latest
of all his inventions — " omniimi ultimo a nobis inventum sit
hoc Multiplicationis Promptuarium " — though he gives it
precedence in the volume, according to his estimate of its
importance. He had previously invented his mode of calcu-

INTRODUCTION. XV

lating with the abacus or chess-board, in the preface of which
he again refers to the origin of all these inventions, — namely,
that it was the constant labour of his Hfe to rend the fetters
with which science was yet subdued. " In the course," says
he, " of devoting every moment of my leisure," — doubtless,
from what he considered his great caUing, the exposition of
the Revelations, — " to the invention of these compendious
methods of calculation, and to the enquiry by what means the
labour and toil of calculation might be removed, besides the
Logarithms, Rhabdologia, the Promptuary of MultipUcation,
and other devices, I hit upon a certain arithmetical table,
which, as it performs the more troublesome operations of
common arithmetic upon an abacus or chess-board, may be
considered an amusement rather than labour ; for, by means
of it, addition, subtraction, multipHcation, division, and even
the extraction of roots, are accomphshed, simply by moving
counters hither and thither upon the board. UnwiUing either
to consign it to obUvion, or to pubUsh so smaU a matter by
itself, I have prefixed it to my Rhabdologia, in addition to the
Promptuary, for the benefit of the studious — and the criticism
of the learned."

That Napier composed his Rhabdologia after the year 1614,
when he pubUshed his Canon of the Logarithms, is also indi-

XVi INTRODUCTION.

cated by the fact, that in the minor work (Lib. I. cap. 2) he
selects, as a numerical example, " Annus Domini 1615." Now,
supposing the Rhabdologia an invention of earlier date than
the Logarithms (which latter, as will presently appear, must
be referred to some time in the previous century), the year of
our Lord 1615, when thus selected for the example, was a year
yet to come ; and it is not likely that such an example would
have occurred to him, in place of a year either present or past.
Probably the year selected was that in which Napier was writing
at the time, a supposition perfectly consistent with his state-
ment, in the letter to the Chancellor, that this minor invention
was for the benefit of those who might be disinclined to use the
Logarithms, and that it had become well known, and even been
carried abroad, before he was induced to publish it. For 1615
was the year after the publication of the Logarithms, and
the Rhabdologia was not printed until 1617.

There is only one circumstance which seems to interfere
with this chronology ; but it is easily explained. In the fifth
chapter of the second book of his Arithmetic, when expound-
ing various compendious methods of multiplication and division,
Napier adds, " Sive omnium facillime per ossa Rhabdologiae
nostne." This might be supposed to afford decisive evidence
that the Rhabdologia was composed before the book of Arith-

INTRODUCTION. xvii

metic. It must be observed, however, that the expressions
quoted are written on the margin of Robert Napier's manu-
script, and there is a mark in the text where the note should
be inserted, — a direction complied with in printing the present
volume (p. 42). Probably this note was an addition of the
transcriber's ; or perhaps Napier himself had subsequently
added it to his MS., after the invention of the calculating rods
had occurred. If the note be contemporary with the composi-
tion of the treatise, then undoubtedly the latter is of a sub-
sequent date to his Rhabdologia. But there is another circum-
stance which seems to render this supposition totally out of the
question. In Napier's Arithmetic occurs a genesis of decimal
fractions, and perhaps the earhest on record. But the peculiar
notation of decimals, — ^from which the system derives all its
power as a reciprocation of the Arabic scale, and which Napier
could not have failed to use and comment upon in any notice
of a decimal fraction, when once aware of the expedient, —
is not hinted at in the treatise on Arithmetic. Now, in the
Rhabdologia, there is a section (of the fourth chapter) entitled
" Admonitio pro Decimali Arithmetica," — being his exposition
of that very notation of decimals which is in use at the present
day. Napier had hit upon this admirable expedient when
constructing his Canon of the Logarithms, — the first mathe-
matical work in which the arithmetic of decimal fractions was

XVlll

INTRODUCTION.

developed, and displayed in fuU operation. This circumstance
of itself appears to determine that the composition of his trea-
tise on Arithmetic is prior in date both to the invention of the
Logarithms and of the Rhabdologia.

It is very probable that Napier's progress, in composing
a complete digest of Nmnbers, might have been interrupted,
and that work set aside for the time, in consequence of those
verv operations having suggested his more profound inven-
tions ; but that he composed his Arithmetic and Algebra sub-
sequently to those inventions is most improbable, considering
that the dates of his pubUshed works are 1614 and 1617, and
that he died in the month of April of the latter year. More-
over, it appears that, at the time of his death, Napier was
occupied in bringing to perfection his most elaborate and
beautiful work, the " Constructio Logarithmorum ; " which
treatise, although it did not see the light until two years
after his death, when it was edited by his son Robert, he had
left in a state for pubhcation. This volume, with some pro-
found and original aids to the science of trigonometry added
as a supplement to it, must have occupied all the time which
he devoted to mathematics, (for he was still deeply engaged
with the Apocalypse,) between the date of the pubUcation of
the Canon of Logarithms and the date of his death. The

INTRODUCTION. xix

concluding sentences of his Canon afford so interesting a
view of the last labours of Napier's Ufe, that I may here
quote them from the Enghsh translation of 1616, which he
revised himself : —

" Now, therefore, it hath been sufficiently showed that there
are Logarithms, what they are, and of what use they are ; for
with help of them, we have both demonstratively showed, and
taught by examples of both kinds of trigonometry, that the
arithmetical solution of any geometrical question may most
readily be performed without trouble of multiphcation, division,
or extraction of roots. You have, therefore, the admirable
table of Logarithms that was promised, together with the most
plentiful use thereof, which, if (to you of the learned sort) I by
your letters understand to be acceptable to you, I shall be
encouraged to set forth also the way to make the table. In
the mean time, make use of this short treatise, and give all
praise and glory to God, the high Inventor and Guider of all
good works."

Then foUows an isolated sentence which he terms " Admo-
nitio ;'* and, probably, he refers to his theological labours when,
in that sentence, he speaks of " rerum graviorum cura" having
interfered with the perfecting of his mathematical work : —

XX

INTRODUCTION.

" Seeing that the calculation of this table, which ought to
have been perfected by the labour and pains of many calcula-
tors, has been accomplished by that of one alone, it will not
be surprising if many errors have crept into it. I beseech
you, benevolent readers, pardon these, whether occasioned by
the weariness of calculation, or an oversight of the press ; for
as for myself, the infirm state of my health, and weightier
occupations, have prevented my adding the last finish. But
if I shall understand that the use of this invention proves
acceptable to the learned, I may, perhaps, in a short time
(God wiUing) publish the philosophy of it, and the method
eithec of amending this Canon, or of constructing a new one
upon a more convenient plan ; that thus, through the dili-
gence of many calculators, a Canon, more highly finished and
accurate than the labour of a single individual could accom-
pUsh, may at length see the light. Nothing is perfect at its
birth."

Yet the work had been long and intensely laboured, and
came firom his hands nearly perfect. No sooner had Kepler
perused it, than he addressed his enthusiastic letter to this
^' most iUustrious Baron," as he caUs him,— the mysterious
hennit of the sciences, whose startUng appeal to the great
continental astronomers, issuing from the terra incognita of

INTRODUCTION. xxi

Scotland, had come upon the immortal German like a voice
from another world. Ere Kepler's letter reached Scotland,
Napier was no more. The response came not from " The
Baron of Merchiston," who was no longer to be found in his
old tower on " the Borough-muir," nor in his yet more
romantic retreats in the Lennox and Menteith. Had he been
spared to reply, and to express his dehght at the homage of
Kepler, it might have been said, of this eloquent and electric
sympathy between the distant orbits of Kepler and Napier, —

" Rude Scotia's mountains now have found a tongue ;
Benlomond answers, through his misty shroud,
Back to the joyous Alps, who call to him aloud,"

But, to return to a less poetical view of the matter, Robert
Napier, the transcriber of the manuscripts now pubUshed, was
he who responded to Kepler's ardent request, by pubhshing
his father's " Logarithmorum Canonis Constructio;" and, to
complete this review of the nature and chronological order of
Napier's mathematical studies, I shall here translate the preface
to his posthumous work, which was published in 1619 :—

" Some years ago, my father, of ever venerated memory,
published the use of the Wonderful Canon of Logarithms ; but
the construction and method of generating it, he, for certain

j^ INTRODUCTION.

rcmmin, was unwilling to commit to types, as he mentions upon
the seventh and the last pages of the Logarithms, until he knew
how it was judged of and criticised by those who are versed in
this department of letters. But since his death I have been
ftssured, from undoubted authority, that this new invention is
much thought of by the most able mathematicians, and that
nothing would deHght them more than if the construction of
his Wonderftil Canon, or so much at least as might suffice to
illustrate it, were pubUshed for the benefit of the world.
Although, therefore, it is very manifest to me that the author
had not put his last finish to this httle work, I have done
what in me lay to satisfy their laudable desires, as well as to
afford Some assistance, especially to those who are weak in such
studies, and apt to stick at the very threshold. I doubt not,
however, that this posthumous work would have seen the Hght
in a far more perfect and finished state, if to the author himself,
my dearest father, — who, according to the opinion of the best
judges, possessed among other illustrious gifts this in particular,
that he could exphcate the most difficult matter by some sure
and easy method, and in the fewest words, — God had granted
a longer use of life. You have, then (benevolent reader), the
doctrine of the construction of Logarithms, which here he
calls artificial numbers, — ^for he had this treatise beside him
composed for severaJ years before he invented the word

INTRODUCTION. xxiii

Logarithms, — most copiously unfolded, and their nature, acci-
dences, and various adaptations to their natural numbers, per-
spicuously demonstrated. I have also thought good to subjoin
to the Construction itself a certain appendix, concerning the
method of forming another and more excellent species of
Logarithms, to which the inventor himself alludes in his epistle
prefixed to the Rhabdologia, and in which the Logarithm of
unity is 0. The treatise which comes last is that which, tend-
ing to the utmost perfection of his Logarithmic trigonometry,
was the fruit of his latest toil, — namely, certain very remarkable
propositions for resolving spherical triangles, without the neces-
sity of dividing them into quadrantal or rectangular triangles,
and which are absolutely general. These, indeed, he intended
to have reduced to order, and to have successively demon-
strated, had not death snatched him from us too soon. I have
added some lucubrations upon these propositions, and also upon
the new species of Logarithms, calculated by that most excel-
lent mathematician, Henry Briggs, pubUc professor in London,
who undertook most wiUingly the very severe labour of this
Canon, in consequence of the singular affection that existed
between him and my father of illustrious memory, — ^the method
of construction and explanation of its use being left to the
inventor himself. But now, since he has been called from this
Ufe, the whole burden of the business rests upon the learned

j^y INTRODUCTION.

Brigg8, as if it were his peculiar destiny to adom this Sparta.
In the mean while, reader, enjoy these labours such as they are,
and receive them in good part. Farewell.

" ROBERT NEPER."

From this history of Napier'8 studies, derived chiefly from
himself and his son, the conclusion is inevitable, that the work
now given to the public was composed before his conception
of the Logarithms ; and that that invention, as well as his
subsequent minor inventions, arose out of the very attempt, to
abridge the toils of calculation, which he indicates in the fifth
chapter of the second book of his Arithmetic, " De Multipli-
cationis et Partitionis Compendiis Miscellaneis." In order to
ascertain, then, the age of his Arithmetic and Algebra, we must
determine, as nearly as possible, the period of time when he
formed his conception of the Logarithms.

No one who has taken the trouble to examine Napier's
original works, or who understands any thing of the nature
and construction of Logarithms, can doubt that many years of
labonr must have elapsed, between the first idea of the system
in the inventor*s mind, and its publication, in 1614, accom-
panied with calculated tables of Logarithms. The work pub-
lished after his death had been composed for several years, as

INTRODUCTION. XXV

we are informed by his son, ere Napier had compounded the
term Logarithms, under which denomination, however, they
were first pubhshed. He himself alludes to the long labour
bestowed upon his Canon, and to the interesting fact that his
operose calculations were entirely performed by himself. M.
Biot, in his review and " extrait " of the Memoirs of Napier,
pubUshed in the Journal des Savans for the year 1835, after
passing the highest encomiums upon the genius of Napier's
works, thus speaks of the mechanical labour of the calcula-
tions : —

" Besides the merit of the invention, Napier's tables are a
prodigy of laborious patience. As we reflect upon the time
and toil it must have cost him to calculate all those numbers,
we shudder at the chances there were of his being cut off ere
he had reaUzed his idea, and of its having died with him. It
has been said, and Delambre repeats the observation, that the
last figures of his numbers are inaccurate. This is true ; but
it would have been a more valuable truth to have ascertained
whether the inaccuracy resulted from his method, or from some
error of calculation in applying it. This is what I have done ;
and I have discovered an error of the kind, a very shght error,
in the last term of the second progression, which he forms pre-
paratory to the calculation of his tables. AU the successive

xxvi INTRODUCTION.

Bteps being deduced from this one, the trifling error in question
is thu8 carried on through the calculations. I corrected his
error, and then, adopting his own method, but kbridging the
operations by our more rapid processes of developement, I
calculated the logaritlmi of 5000000, which is the last in
Napier*8 table, and consequently that upon which all the errors
accumulate. I found for its value 6931471.808942 ; whereas,
by the modern series, it ought to be 6931471.805599. Thus
the difference commences at the tenth figure. I calculated,
in like manner, the hyperboUc logarithm, with Napier's
numbers corrected, and found for its value 2.3025850940346,
while, by our present tables, it is 2.3025850929940 ; so
the r^al difference only falls on the ninth decimal. But
this is beyond the range of the tables of CaUet, which are in
daily. use. If Napier had even commanded the services of
a village schoolmaster, to calculate, by the inventor's own
method of subtractions, a geometrical progression stiU slower
than what he used, a desideratum to which he himself caUs
attention, the tables of Briggs, calculated to fourteen deci-
mals, would have possessed no superiority over those of
Napier. His minor inventions are scarcely worth mention-
ing afker this immense invention of Logarithms. That was
sufRcient for the Ufetime, as it is for the fame, of a single
individual.**

INTRODUCTION. XXVll

It is interesting to compare this judgment upon Napier's
labours, pronounced in the nineteenth century from the
highest tribunal of science, with that modest " Admonitio "
of his own, already quoted, wherein he anticipates the chance,
and deprecates the criticism, of casual errors in his unaided
calculations. Biot tells us, that he could only discover " une
petite faute de ce genre, une tres petite faute." We may
assume, then, that Napier's great invention had first occurred
to him many years before its pubhcation. Indeed the foUowing
details, which will serve to determine the chronology both of
the invention of Logarithms and of the treatises now submitted
to the pubhc, afford a curious proof that he was in possession
of the secret, and preparing his pubHcation of the Logarithms,
some time before the close of the sixteenth century.

It is a remarkable fact, not generally known, that Tycho
Brahe obtained some hint of this boon to be conferred
upon science, twenty years before Napier's other avocations,
added to the tedious labour of the calculations and his
own difiidence, permitted him to give his invention to the
world. Sir Archibald Napier of Merchiston, the father of
the great Napier, was Justice-Depute in the reign of Mary
of Scotland. His coUeague in that ofiice was Sir Thomas
Craig of Riccarton, celebrated for his work on Feudal law.

XXviii INTRODUCTION.

The third son of the Feudist was John Craig, afterwards
Dr John Craig, physician to James VI. ; and he was also the
friend of John Napier, the son of his father s coUeague in
oflBce. But there was another, and perhaps the stronger
tie. Young Craig was much given to mathematical studies.
There is one record, indeed, of his success in those pur-
suits, which of itself is suflficient to distinguish his memory
in that respect, — a record rarely met with in his own
country, and still seldomer perused. I allude to a small
volume of Latin epistles, printed at Brunswick in the year
1737, and dedicated by their collector, " Kud. Aug. Nol-
tenius," to the Duke of Brunswick. The three first letters
in thi* coUection are from Dr John Craig to Tycho Brahe,
and they prove that he was upon the most friendly and con-
fidential footing with the Danish astronomer. He addresses
Tycho as his " honoured friend," and signs himself, " your most
affectionate John Craig, Doctor of Philosophy and Medicine."
The first of these letters (which are written in Latin) com-
mences with the fact, that, " About the beginning of last
winter that illustrious man, Sir William Stewart, delivered to
me your letter, and the book which you sent me." The date
of this letter is not given ; but, in the Library of the Univer-
sity of Edinburgh, there is a mathematical work by Tycho
Brahe, which bears, upon the first blank leaf, a manuscript

INTRODUCTION. xxix

sentence in Latin to the foUowing effect : " To Doctor John
Craig of Edinburgh, in Scotland, a most illustrious man,
highly gifted with various and excellent learning, Professor of
Medicine, and exceedingly skilled in the Mathematics, Tycho
Brahe hath sent this gift, and with his own hand hath written
this at Uraniburg, 2d November 1588."

It appears from contemporary records, that, in the month
of August 1588, Sir WilHam Stewart, commanding the royal
guard of Scotland, was sent to Denmark to arrange the preh-
minaries of King James's marriage, and that he returned to
Edinburgh upon the 15th of November 1588. There can be
no doubt that the volume above-mentioned is that referred to
in Craig's epistle to Tycho. Neither can it be doubted that
this was Dr John Craig, third son of Sir Thomas Craig of
Riccarton, and physician to James VI. He was raised to his
highest post in the royal household by a gift from that monarch,
recorded in the Foedera, " dilecto nobis Johanni Craigio
in medicinis doctori, oflficium et locum ordinarii, et primarii
medici nostri," dated at Westminster, 20th June, 1603. This
was the friend of Tycho Brahe and of Napier. The Danish
astronomer transmitted his present to John Craig, from his
palace of Uraniburg, with the air of the Monarch of Science.
Napier, from his old tower of Merchiston, when writing, in

XXX INTRODUCTION.

ihe year 1608, to his own son, who was gentleman of the bed-
chamber to King James in England, thus remembers the
King^s physician : — " Ye sall make my commendatiouns to
Doctor Craig."

The facts now stated serve to throw light upon a sentence,
respecting the invention of Logarithms, which occurs in one of
Kepler'8 letters, — a sentence which has been little noticed, and
never rightly understood. Kepler, when writing to his favourite
correspondent Petrus Cugerus (a mathematician of Dantzic,
and the master of Helvehus) upon the subject of the economy
of the heavenly bodies, and after reveling in his deepest calcu-
lations^ declares that, of aU the methods of calculation in aid
of astronomy, nothing excels the invention of Napier ; and yet
(he adds), even so early as the year 1594, a certain Scotchman,
in a letter written to Tycho, held out some promise of that
Wonderftil Canon. The precise words of Kepler's letter are,
" Nihil autem supra Neperianam rationem esse puto : etsi,
quidem, Scotus quidam, literis ad Tychonem A. cioiDXCiv.
scriptis, jara spem fecit Canonis illius Mirifici." Kepler was
the pupil of Tycho, and joined him as such after the reverse of
fortune which expelled the illustrious Dane from his palace of
Uraniburg. This aecounts for Kepler's having obtained some
partial knowledge of the letter to whieh he alludes. But who

INTRODUCTION. XXXI

was the " certain Scotchman" from whom Tycho received
this important communication before the year 1594, — a date
so precisely given by Kepler, that we cannot doubt its
accuracy ?

John Craig had long intended to pay a visit to Tycho
Brahe. This appears by his letter of the year 1589, ah*eady
quoted, in which he states that five years before he had made
an attempt to reach Uraniburg, but had been baffled by
storms and the inhospitable rocks of Norway ; and that ever
since, being more and more attracted by the accounts, brought
by ambassadors and others, of Tycho's fame, and the magni-
ficence of his observatories, he had been longing to visit him.
In the year 1590, James VI., the patron of Craig, spent some
days at Uraniburg, before returning to Scotland from his
matrimonial expedition. It cannot be doubted that James's
physician, who was long about this monarch's person in a
medical capacity, and eventually at the head of his medical
staff, would seize the prtfpitious opportunity, of the progress
of his royal master, to visit his friend ; and it is not unlikely
that Craig himself had suggested or encouraged the visit of
his royal master to Uraniburg. Dr Craig, it is most pro-
bable, was the " certain Scotchman" to whom Kepler alludes,
and who, in the year 1594, had written a promise of the

XXxii INTRODUCTION.

Logarithms to Tycho Brahe. For after his visit to Tycho,
in the year 1690, the person in Scotland, to whom Craig would
mo8t eagerly unfold the wonders of Uraniburg, was Napier of
Merchiston.

Frederick II. of Denmark, the munificent patron of Tycho,
had established that philosopher on the island of Huen, situated
at the mouth of the Baltic, adding honours and revenues, and
every aid and encouragement which the most ardent astro-
nomer could desire. Upon the 8th of August, 1576, the first
stone of the far-famed castle or palace of Uraniburg was laid
in Tycho's principality. It was a vast quadrangle, the dimen-
sions heing sixty feet every way, and flanked with lofty towers
thirty-two feet in diameter, the observatories of this palace of
science. Tycho is also said to have fitted it up with certain
mysterious tubes, and other telegraphic contrivances, which
enabled him to communicate with his domestics as if by magic,
and obtain secret knowledge of his many visitors long before
their arrival.

In the year 1590, that in which King James visited Tycho,
Napier^s fertile genius, unaided by the encouragement of royal
patronage, was teeming with various discoveries in mechanical
science (besides his speculations in the science of Numbers),

INTRODUCTION. xxxiii

which, Hke Archimedes of old, he intended should be appHed
for the defence of the island against foreign invasion. A
report made to Napier by his friend Dr Craig, of all he
had seen and heard at Uraniburg, when there with his
Majesty, would perfectly account for the following sentence of
Napier's admonitory letter to King James (on the subject of
his Majesty's supposed incHnation to Popery), dated from
Merchiston, 29th January, 1593 : — " For let not your Majesty
doubt but that there are within your realm, as well as in other
countries, godly and good ingynes, versed and exercised in aH
manner of honest science, and godly discipHne, who, by your
Majesty's instigation, might yield forth works and fruits
worthy of memory, which otherwise, lacking some mighty
Mgecenas to incourage them, may perchance be buried with
eternal silence."

His conversations with Dr Craig might suggest to Napier,
not merely this hint to King James, that his Majesty should
patronise science in Scotland, but the transmission of a hint
to Tycho himself on the subject of astronomical calculation.
If the King of Denmark, when he invested Tycho with some-
thing Hke eastern splendour, could have added the power that
stiU lay hid in Arabic numbers, a false system of the world
would not have been re-estabHshed at Huen. It was in the

Xxxiv INTRODUCTION.

metns of astronomical caJciilation that the science of Tycho

Brahe was most deficient. Now Kepler tells us, that even so

early as in the year 1594, (the year following that in which

Napier^s letter to King James is dated,) a certain Scotchman

had written to Tycho some promise of the Logarithms. After

the facts referred to, it can scarcely be doubted that this

correspondent from Scotland was none other than Tycho's

old friend and correspondent there, the learned physician of

King James, and also the friend of Napier, with whom he

could not fail to have often discussed the subject of the royal

visit to Uraniburg. And this appears to be placed beyond

all doubt, when, to what has been already stated, we add

the foilowing anecdote, somewhat imperfectly recorded by

Anthony-a-Wood.

That indefatigable and amusing collector of literary gossip
thus narrates it, in the Athenae Oxonienses : — " It must now
be known that one Dr Craig, a Scotchman, coming out of
Denmark into his own country, called upon Joh. Neper, Baron
of Marcheston, near Edinburgh, and told him, among other
discourses, of a new invention in Denmark, by Longomon-
tanus, as 'tis said, to save the tedious multiplication and divi-
sion in astronomical calculations. Neper being soHcitous to
know further of him concerning this matter, he could give no

INTRODUCTION. xxxv

other account of it than that it was by proportional numbers.
Which hint Neper taking, he desired him at his return to call
upon him again. Craig, after some weeks had passed, did so,
and Neper then shewed him a rude draught of what he called
Canon MirabiUs Logarithmorum. Which draught, with some
alterations, he printing in 1614, it came forthwith into the
hands of our author Briggs, and into those of Will. Oughtred,
from whom the relation of this matter came."

Any one at all conversant with the history of science, and
the nature of the invention of Logarithms, will at once per-
ceive, that, whatever foundation in fact this anecdote may
have, it is here inaccurately and ignorantly told. Longomon-
tanus was the pupil and assistant of Tycho at Uraniburg, and
a most distinguished mathematician. He, in common with
many others, was well acquainted with a principle of numerical
progression, the extraordinary generahzation of which by
Napier is that which constitutes the invention of Logarithms.
Archimedes had first observed and speculated upon such pro-
gressions, but without discovering the Logarithms. It could not
possibly be, that a hint from Longomontanus had suggested
to Napier his great invention ; for if a hint of the kind could
have urged any human intellect thus rapidly upon the concep-
tion of the Logarithms themselves, that hint had arisen in the

Xjutvi INTRODUCTION.

school of Alexandria, was submerged in the middle ages, and
rose again with the letters of Greece,— a hint which Tycho
had, which StifeUius, Byrgius, Longomontanus, and Kepler
himself had ; yet no more was made of it after the revival of
letters, than had been by Archimedes before their fall. Kepler
too, informs us, that in the year 1594, something regarding'
the generalization of this numerical principle had actually been
reported in a letter to Tycho ; yet the secret was still undis-
covered until Napier published his Canon in 1614. In a
letter, dated llth March, 1618, to his friend Schikhart,
Kepler, after descanting upon the various diflficulties and
resources of trigonometry, exclaims, — " A Scotish Baron has
startcd up, his name I cannot remember, but he has put fortli
some wonderfiil mode by which all necessity of multiplications
and divisions is commuted to mere additions and subtractions,
nor does he make any use of a table of sines ; still, however,
he requires a canon of tangents, and the variety, frequency, and
difficulty of additions and subtractions, in some cases exceed the
labour of multiplication and division." This was the first crude
and inaccurate idea formed by Kepler of the work which he had
not yet studied ; and abeady the Scotish Baron whose name
he could not remember was in his grave ! But of this fact
Kepler was not aware even on the 28th July, 1619, when he
thus addressed Napier himself : —

I^

INTRODUCTION. XXXvii

" But the chief cause that impeded my progress this year,
in framing the Rudolphine Tables, was an entirely new, but
happy calamity, which has befallen a part of the Tables I had
long ago completed. I mean, most illustrious Baron, that
book of thine, which, pubHshed at Edinburgh in Scotland five
years ago, I first saw at Prague the year before last. At that
time I had not leisure to study it ; but last year, having met
with a Kttle book by Benjamin Ursin (long my famihar, and
now astronomer to the Margrave), wherein he briefly gives the
marrow of the matter, extracted from your own work, I was
awakened to its merits. Scarcely, indeed, had I made trial of
it, in a single example, when I became aware, to my great

dehght, that you had generahzed a certain play of Numbers,
which I myself, in a very minute degree, had practically
adopted for many years, and had proposed to incorporate with
my Tables ; especially in the matter of parallaxes, and in the
minutes of duration and delay in echpses ; of which method
this very Ephemeris exhibits an example. I knew, indeed, that
this method of mine was only apphcable in the single case of
an arc difFering in no sensible degree from a straight hne.
But of this I was ignorant, that, from the excesses of the
secants, Logarithms could be generated, thus rendering the
method universal throughout any extent of arc. Then indeed
my mind, above all things, was eager to ascertain, whether,

XZXVUl

INTRODUCTION.

in this little book of Ursin's, these Logarithms had been
aocurately investigated CalHng to my aid, therefore, Janus
Gringalletus Sabaudus, my familiar, I ordered him to subtract
the thousandth part of the whole sine,— again, to subtract the
thousandth part of that residue, and to repeat this operation
more than two thousand times, until there remained about the
tenth part of the whole sine ; but of the sine from which a
thousandth part had been subtracted, I computed the loga^
rithm with the greatest care, beginning from the unit of that
division which Pitiscus most frequently uses, namely, the duo-
decimal. The logarithm thus computed, I arranged uniformly
with the remainders of all the subtractions. In this manner,
I asc^ained that these logarithms were as nearly perfect as
possible, although a few errors had crept in, either of the press,
or in that minute distribution of the greater logarithms about
the beginning of the quadrant. I mention all this to you by
the way, in order that you may understand how gratifying it
would be, to me at least (and I should think to others), if you
would put the world in possession of the methods by which you
proceeded. Of these, I make no doubt, you have many, and
most ingenious, at your command intuitively. And so the
promise held out by you on the 57th page of your work," (to
put the world in possession of his methods of constructing the
Logarithms, should he understand that the invention was

INTRODUCTION. XXxix

approved of by men of science,) " has fallen due to the public.
And now let us grapple with your Tables." .

This interesting letter, written after he to whom it was
addressed had been dead for nearly two years, aflfbrds the most
complete refutation of that part of the anecdote, in the Athense
Oxonienses, which seems to impute Napier's invention to a
sudden thought suggested by a hint from Kepler's intimate
friend and companion Longomontanus. Kepler himself, in
this letter, refers the invention to its true source, namely,
Napier's surpassing command of the science of Numbers.
Kepler, whose own immortal discoveries in Astronomy obtained
for him the daring title of Legislator of the Heavens, bowed
at the shrine of the Logarithms, and at once, and without
reserve, acknowledged Napier as his master in Logistic. A
quaint indication, of his estimate of Napier's invention, is also
to be discovered in the frontispiece to those Rudolphine Tables
of which he speaks in his letter. The telescope of Galileo, the
eUiptical orbit of a planet, the system of Copernicus, and a
female figure with the Napierian Logarithm of half the radius
of a circle arranged as a glory round her head, are there
dehneated as figurative of the mighty impulses which Astro-
nomy had received in those days. But that the anecdote of
Anthony-a-Wood has some foundation in fact is obvious, when

_j INTRODUCTION.

we compare it with what has been previously stated. The Doctor
Craig therein mentioned is undoubtedly Napier's friend, the
physician of King James ; and the " coming out of Denmark
into his own country," refers to his return with his Majesty
in 1590. Moreover, that Craig had some conversation with
his friend the Baron of Merchiston on the subject of astrono-
mical calculations, and that the result was some communication
on the subject of the Logarithms, seems to be verified by the
independent evidence affbrded by Kepler's other letter, where
he says that a certain Scotchman, in a letter to Tycho written
in the year 1594, already held out some hopes of that won-
derful Canon ; a fact which Kepler had subsequently become
aware of, probably from his inspection of Tycho's scientific
papers and correspondence. Thus, by a very curious chain
of evidence, it seems proved, that Napier had made some
advance in his construction of the Logarithms so early as
1594, and that he had set himself deliberately to the task,
after the difficulties which harassed the tlirone of iVstronomy
at Uraniburg (in consequence of the crudeness of numerical
science) had been reported to him in 1590. But, as Kepler
himself observes, the Inventor of Logarithms must have
previously commanded many and most ingenious numerical
resources. He must have been far in advance of his age in a
profound knowledge of the mysterious play of Numbers. His

INTRODUCTION. xli

mind must have been thoroughly and deeply imbued with
the Logistic art, and analytical power, ere he proposed to him-
self the invention of that great lever of science. Now it is in
the Latin treatises, which these observations preface, that we
may trace Napier's command of Numbers, before he had
formed a conception of the Logarithms. And thus the pre-
sent work may be said to afford a most interesting chapter in
the history of the progress of Science. A digest of the
whole art of Logistic, equal, in so far as the then existing
numerical notation admits of the comparison, to Euler's in
modern times, composed prior to the year 1594, amid the
storms of faction which then distracted Scotland, was an
achievement worthy of being crowned by the invention of
Logarithms, for the natural discovery of which, in the gradual
development of algebraic resources, the analytic art was at the
time an age too young.

Another circumstance may here be mentioned which also
affords evidence, that, many years prior to the pubhcation of
the Logarithms, Napier's fame as a mathematician was esta-
bhshed with the learned of his own countrymen, and that he
was the person to whom Dr Craig would be most eager to
converse on the subject of his visit to Tycho Brahe. The
celebrated lawyer, and learned and accompKshed author, Sir

/

Xlii INTRODUCTION.

John Skene of Cuirie Hill, when, in the course of preparing
his treatise " De Verborum Significatione" (first published in
the year 1597), he came to the word " particata vel perticata
teiraB," which he explains, " from the French word ' perche,'
meikle used in the English lawes, ane ruid of land," adds this
sentence : — " But it is necessare that the measurers of land
called landimers, in Latin ' agrimensores,' observe and keep
ane just relation betwixt the length and the breadth of the
measures quhilk they use in measuring of lands, quhairanent I
find na mention in the lawes and register of this realme, albeit
ane ordinance thereanent be made by King Edward the First,
King of England, the 33d yeir of his reigne ; and because the
knawledge of this matter is very necessare in measuring of
lands dayly used in this realme, I thought good to propone
certaine questions to John Naper, fear of Merchistoun, — ane
gentleman of singular judgement and learning, especially in
the mathematical sciences, — the tenour quhairof, and his
answers made thereto, foUowis." Napier's answers are of con-
siderable length. and given with his characteristic simplicity
and power.

If the composition of the treatises now published is to
be referred to a date prior to the year 1594, which seems to
be placed beyond doubt by what has been stated, they are

INTRODUCTION. xliii

interesting not merely as the works of the Inventor of
Logarithms, but as being the very first of the kind in Scot-
land, and among the first systematic works on Numbers
after the revival of letters in Europe. The sixteenth cen-
tury was the rudest period of algebraic science in Europe.
Leonardo of Pisa, indeed, composed his work before the inven-
tion of printing, and early in the thirteenth century ; but this
had been lost sight of, and was not known for more than a
century after Napier's death, when the manuscript was disco-
vered at Florence. The first printed work on the subject was
that of Lucas de Burgo, from whom is generally dated the
decided dawn of Algebra in Europe. De Burgo's principal
work was printed about the year 1494. The second printed
work upon Arithmetic and Algebra appeared in 1539. This
was a work of the great but eccentric Cardan, of whom it is
affirmed by ScaUger that he was so devoted to astrology as
actually to starve himself to death, that his own astrological
prediction might be fulfilled, — a very equivocal illustration of
his favourite science. Cardan died in the year 1575. Germany
produced one or two mathematicians, who, at the same time
that Cardan wrote, gave a more decided impulse to Numbers.
Hitherto nothing had been added to that recondite science,
since the introduction of the Arabic numerals, and the rude
and imperfect symbols of Burgo's Algebra, except in the

jUy introduction.

theory of equations, which had received a great extension
from Tartalea and Cardan. The defect which materially
inipeded the system was in notation, the mainspring of numer-
ical science. In the year 1544, Michael Stifellius, a Lutheran
clergyman, published at Nuremberg his Arithmetica Integra,
a very original Latin treatise on Arithmetic and Algebra,
wherein he viewed numerical quantities, and their combina-
tions, closely and ingeniously, and gave an impulse to Algebra
by improving its notation. He was the first to introduce the
signs -|- and — for plus and minus, and also the character ^/
(derived from the letter R), to denote the radix or root.
Moreover, he entered systematically into the consideration of
arithrtetical and geometrical progressions, pointing out what
may now be termed the logarithmic properties of a correspond-
ing series of the powers of a given number, and the exponents
of those powers, which latter term he uses. But in this spe-
culation he had formed no conception of the possibility of
changing the infinite series of natural numbers from an arith-
metical to a geometrical progression, and then of generating a
corresponding arithmetical progression.

In the year 1552 appeared the first treatise upon Arith-
metic and Algebra in the English language. Its author was
tbe unfortunate Robert Recorde, a mathematician of un-

INTRODUCTION. xlv

doubted genius, but so little appreciated that he was allowed
to Hnger out his last days in the Fleet, where he was im-
prisoned for debt, and where he died about the year 1558.
His works are curious and original, but rude and puerile
in their style. They are generally in the form of a dialogue
between a Master and Scholar, and under such quaint titles,
as " The Pathway to Knowledge " — " The Ground of
Arts"— " The Castle of Knowledge"— " The Whetstone of
Wit." Napier's style of communicating even his elementary
rules is clear, simple, and philosophical. His son tells us, that,
" according to the opinion of the best judges, my dearest
father possessed, amongst other great endowments, this in
particular, that he could expHcate the most difficult matter by
some sure and easy method, and in the fewest words." Ac-
cordingly, he never indulged in such facetise as the foUowing,
which occurs in Recorde's Arithmetic, a work, nevertheless,
of original genius : — " Master. ' Exclude number, and answer
this question — How many years old are you ?' — Scholar.
' Mum.' — Master. ' So that, if number want, you answer all
by mummes. How many miles to London ?' — Scholar. ' A
poak fuU of plums.' — Master. ' If number be lacking, it maketh
men dumb ; so that, to most questions, they must answer
mum. What call you the science you desire so greatly ?' —
Scholar. ' Some caU it Arsemetrick, and some Augrime.' —

jflyl INTRODUCTION.

Biaster. ' Both names are corruptly written ; Arsemetrick
for Arithmetic, as the Greeks call it ; and Augrime for
Algorisme, as the Arabians sound it.' "

Recorde was ingenious and inventive, and is remarkable,
in the history of Algebra (which owes so much to successive
inventions in the art of notation), for having added to its
characters the sign of equaUty. Mr Babbage had overlooked
the fact. That distinguished mathematician, in his history
of notation, observes, " It is a curious circumstance that the
symbol which now represents equality was first used to denote
subtraction, in which sense it was employed by Albert Girard,
and that a word signifying equality was always used instead,
until the time of Harriot." The works of Girard and Harriot
did not appear until the seventeenth century was far advanced,
and long after Napier's death. Napier's book of Arithmetic,
as already shown, must be referred to a date prior to the year
1.594 ; and it will presently appear that his book of Algebra is
of a date still earHer. If Mr Babbage's observation were
historically accurate, we might claim for Napier the merit of
having hit upon this ingenious device, in his unpublished work,
a century before the time of Harriot. For, it will be observed,
he adopts it in his chapter of equations, and defines it in these
terms : — " Betwixt the parts of an equation that are equal to

INTRODUCTION. xlvii

each other, a double line is interposed, which is the sign of
equaHty ; thus 1 1^=7, which is pronounced, one thing equal
to seven." The fact is, however, that this notation was already
more or less in use ; and it is to Recorde that the merit of the
first idea is due ; for, in his work, first pubHshed in 1552, he
says, " And to avoid the tedious repetition of these woordes,
' is equal to,' I wiH sette, as I doe often in woorke use, a paire
of paraUeles, or gemowe Hnes of ane length, thus =, because
noe two thynges can be moare equalle."

In France, the celebrated Ramus wrote an elementary
treatise on Arithmetic and Algebra, about the year 1560 ;
but he left the science as he found it. Raphael BombeUi,
whose Algebra was pubHshed at Bologna in the year 1572, in
ItaHan, wrote more elaborately and profoundly, but did not
add any thing of consequence to the labours of his predecessors.
The first that can be said to have done so, between the time
of StifeUius and of Napier, was Simon Stevinus of Bruges,
who pubHshed La Practique d'Arithmetique about the year
1582. He afterwards put forth other works upon Arithmetic
and Algebra, along with a translation of some books of Dio-
phantus ; in aU of which he evinced a remarkable genius for his
subject. Algebraic notation received at his hands another of
those impulses by which it has so graduaUy reached its present

jjyiii INTRODUCTION.

power ; and he appears to have been the first who expressly
mooted the doctrine of a decimal division of unit, which idea,
however, remained to be developed and practically appHed as
a system by Napier.

A contemporary of Napier's was Vieta, whose name
reflects lustre upon France. He generaUzed the language of
Algebra, by employing letters to denote known as well as
unknown quantities, and he extended the theory of equations.
It is not at all likely that Napier ever saw any of his treatises,
which were only first coUected into one volume by Schooten,
in 1646. AU the other conspicuous treatises, illustrative of
the progress of numerical science, are subsequent to the death
of Napier. At some period of his life he had perused the
work of Stevinus, as he mentions this author in the Rhabdo-
logia ; but considering how very few works of the kind existed
at the time when Napier must have composed his numerical
treatises, how slowly books were then spread abroad, and that
literary communication between Scotland and the Continent
was 80 sHght as to leave Kepler in ignorance of the death of
the Inventor of Logarithms two years after that event, it is
not to be supposed that Napier had at his command even those
scanty stores of information which other countries could afford,
on the abstruse subjects to which he was devoted. The most

INTRODUCTION. xlix

accurate chronology of his treatises, so far as can be ascer-
tained, would seem to be between the pubHcations of Stevinus
of Germany, in 1582, and those of Vieta of France, about the
commencement of the next century. Of all his predecessors
and contemporaries, however, Napier was the one destined to
create the greatest revolution in Numbers. This fact alone
must render the perusal of his early and preparatory labours
very interesting to all who are capable of appreciating his
great invention. Countries the most distinguished in Europe,
for men of science, had produced in that recondite path the
few remarkable men so briefly and imperfectly noticed above.
But even the gigantic Kepler (on his own authority we state
it) had struggled in vain against the spell that yet bound the
Arabic system. From Archimedes to Kepler, not one of the
victorious in the field of science had struck a blow sufficient
to extricate the best wing of mathematics. Until Napier
arose, the Arabic scale remained undeveloped. The name
of Recorde is barely sufficient to give England a place in the
previous history of the progress of science ; and as for Scot-
land, when Napier was acquiring for his country the fame of
one of the greatest impulses which human knowledge has
received, it was distinguished by mists which science had not
penetrated, and by the Douglas wars, whose baronial leaders
knew Httle of the denary system beyond their mail-clad hands.

1 INTRODUCTION.

Napier, in the first chapter of his Arithmetic, refers to
what he terms Geometrical Logistic, as forming the subject
of his third book, and to Algebra, as being the subject of a
fourth book. It would appear, however, that his Algebra,
80 far as " orderlie sett doun," is an earlier production than
either his Arithmetic or the fragment of Geometrical Logistic.
This is manifest from several circumstances : — 1. It is entitled
" The Algebra of John Naper, Baron of Merchistoun ;" but
not Liber Quartus, in correspondence with the other books.
2. Arithmetic is referred to in it ; but there is no reference to
his own book of Arithmetic, as there would have been, accord-
ing to his practice throughout the rest of the manuscript, had
that been previously composed. 3. The treatise of Algebra is
itself divided into two books ; and, while there is a systematic
reference to its component parts, there is no reference to any
previous books. 4. Napier adopts in his Algebra the nominar
tion and notation which had been introduced before his time ;
whereas, in liis Arithmetic and Geometrical Logistic, he
adopts and expounds a peculiar nomination and notation of
his own, applicable to the arithmetic of Surds, and whereby
he proposes to supersede that with which he operates in his
treatise of Algebra. There can be little doubt, therefore, that
his Algebra is a work of still earher date than the other books ;
and these, as has been seen, were prior in date to his conception

INTRODUCTION. U

of the Logarithms, which was some time before the year 1594.
Yet there is no appearance of crude or hasty composition in
his earHest work. It is stamped with the same characteristics —
clear and simple exposition, profound and original views, and
symmetrical arrangement — which eminently distinguish all his
compositions.

Having ascertained the most probable chronology of the
treatises now printed, as being the occupations of a period
embracing a course of years prior to 1594, we must compare
the fact with Napier's time of Hfe. He was born at Merchis-
ton in the year 1550 ; so that he was only of age in 1571, the
year of his marriage in Scotland. Even at this early period
of his Hfe, he must have been far advanced in his abstruse
studies, in order to have obtained that complete command
of Logistic, — a term he uses to denote the whole analytic
art, — which had enabled him, prior to the year 1594, to
digest every department of numerical science into a more
comprehensive and perfect institute of the subject than had
yet appeared. Napier himself has recorded an anecdote which
illustrates the extraordinary precocity of his genius ; and, while
referring to it, I may add a few particulars relative to his early
studies. It appears, from the original records of the University
of St Andrews, that Napier was incorporated in the CoUege

lii INTRODUCTION.

of St Salvator in the year 1663. But there is an earlier
and very curious notice, in reference to his education. His
mother's brother was that celebrated character, Adam Both-
well, the first reformed Bishop of Orkney ; the same who
performed the marriage ceremony between Queen Mary and
the Earl of Bothwell, and who afterwards anointed and
crowned the infant James VI. This prelate concludes a
letter (still preserved in the Napier charter-chest) on the
subject of his own private affairs, addressed " To his Bruder
the Laird off Merchistoun in Loudeanne," Napier's father,
with the following remarkable and prophetic sentence : —

" 'I pray you, schir, to send your sone Jhone to the
schuyllis ; oyer to France or Flandaris ; for he can leyr na
guid at hame, nor get na proffeit in this maist perullous
worlde, — ^that he may be savet in it, — ^that he may do frendis
efter honour and proffeit, — as I dout not but he will : quhem,
with you and the remanent of our successione, and my sister
your pairte, God mot preserve eternalle. At the yairdis in
Kirkwall, this v day of December, the yeir of God 1560, be
" Your Bruder at powair,

Adame, Bischopp off Orknay."

Napier, at the date of this letter, (in which will be

INTRODUCTION. liii

observed a promiscuous use of the Roman and the Arabic
notation), was ten years of age, and his pubHc education
had not commenced. His father was only about sixteen years
his senior, which probably induced the Bishop to tender his
advice. But it was not until 1563 that young Napier matri-
culated at St Andrews ; and how soon his genius there im-
pelled him to the deepest speculations, we learn from him-
self. In his address " To the Godly and Christian reader,"
prefixed to his Commentaries on the Apocalypse, pubHshed
in 1593, he tells us : — " In my tender yeares and barneage
in Sanct Androis, at the schooles, having, on the one part,
contracted a loving familiaritie with a certaine gentleman,
a Papist, — and, on the other part, being attentive to the
sermons of that worthy man of God, Maister Christopher
Goodman, teaching upon the Apocalyps, I was so moved in
admiration against the bhndness of Papists, that could not
most evidently see their seven-hiUed citie Bome painted out
there so lively by Saint John, as the mother of all spiritual
whoredom, that not onely burstit I out in continual reasoning
against my said famihar, but also, from thenceforth, I deter-
mined with myself (by the assistance of God's Spirit) to
employ my studie and diligence to search out the remanent
mysteries of that holy book, — as to this houre (praised be the
Lorde!) I have bin doing at al such times as conveniently I

liy INTRODUCTION.

might have occasion." It is a curious trait of the early power ..
of his mind, that, when only fourteen years of age, he should
have listened so intensely to an exposition of the Apocalypse
from the pulpit, bursting forth afterwards in disputation with
his papistical friend and companion, until he conceived the
daring project of leaving not a mystery of prophecy undis-
covered, — a project eventually realized, as he supposed, by
those profound but fruitless speculations, in which he has been
foUowed by Mede, Sir Isaac Newton, and a host of moderns,
who have added nothing to his labours. The anecdote not
only proves the precocity of his genius, but indicates the
immediate cause of his very early attraction to that profound
study of numerical science, which must have been co-extensive
with the progress of his very learned and long laboured work,
the " Plain Discovery of the whole Revelation of St John."
His varied illustrations of such propositions as, for instance,
that " the forty-two months, a thousand and two hundred
and threescore prophetical days, three great days and a half,
and a time, times, and half a time, mentioned in Daniel and
the Revelation, are all one date," prove that he was deeply
versed in ancient chronology and numbers. Moreover, in
attempting to expound the mystery of the name and number
666, he evinces a knowledge of the numeral system of the
Greeks, and that they worked arithmetically with the letters of

INTRODUCTION. Iv

their alphabet, instead of the Arabie or Indian notation now
in use. It must be observed also, that the arrangement and
structure of his theological work is mathematical.

There is reason to beheve that Napier had made some
researches abroad, relative to the history of the Arabic notar
tion, when a very young man. In his Memoirs, I had ventured
the hypothesis that he must have been abroad for a time after
his studies at St Andrews. This was founded on the facts,
that, as appears from the records of the University, he did not
remain at St Salvator's long enough to take a degree, or to
complete the usual course of studies there ; and that, from the
terms of a letter of the Bishop of Orkney to the Laird of
Merchiston in the year 1668, young Napier seems to have been
absent from home of that date. Besides, the almost invariable
custom of Scotland then was, for all young men of family, having
any pretensions to a learned education, to complete their studies
on the Continent. Napier's Commentaries on the Apocalypse
prove that he was a most accompUshed scholar, that he pos-
sessed a general knowledge of the arts and sciences, and a
great command of the learned and continental languages.
Under these circumstances, it is more difficult to beheve that
he had never quitted his own rude country, than that he had
obtained the usual advantage of spending some years of his

hri INTRODUCTION.

youth in France, Italy, and the Low Countries; more especially,
88 this very plan of his education had been earnestly recom-
mended by his uncle, the Bishop of Orkney. In coming to
this natural and indeed inevitable conclusion, when writing his
Memoirs, I was not aware of a very interesting fact, subse-
quently mentioned to the Royal Asiatic Society by the Right
Honourable Sir Alexander Johnston, (a Hneal descendant of
the Inventor of Logarithms,) as chairman of the Committee of
Correspondence. At the anniversary meeting of the Society
held on the 9th of May, 1835, Sir Alexander, in the course of
an address, for whose varied information he received the thanks
of the Society, took occasion to communicate the following
facts rfelative to Napier, and the curious result of a projected
Hfe of him by Lord Napier, which I have extracted from
the proceedings of the learned body to whom they were
addressed : —

" The province of Madura again became an object of
literary interest in the eighteenth century, in consequence of
my grandfather, the fifth Lord Napier of- Merchiston, having
determined to write the life of his ancestor, John Napier of
Merchiston, and to prefix to it a history of the knowledge
which the people of India had of mathematics. It appearing
by John Napier's papers, that he had, from the information he

INTRODUCTION. Ivii

obtained during his travels, adopted the opinion, that numerals
had first been discovered by the college of Madura, and that
they had been introduced from India by the Arabs into Spain,
and into other parts of Europe, Lord Napier was anxious to
examine the sources from whence John Napier had derived his
information upon this subject ; and,when he himself was abroad,
he visited Venice and other places in Italy, in which he thought it
was Kkely he should find an account of the information coUected
by the members of the Jesuit mission at Madura, upon this and
other parts of Hindoo science. Having been successful in
obtaining some interesting documents relative to the object of
his researches, he returned to Scotland, and submitted them to
the then Mr Mackenzie (afterwards Colonel Mackenzie), who
had been recommended to him by Lord Seaforth,*as a young
man who had devoted himself to the study of mathematics.
Lord Napier died before he had completed his hfe of John
Napier ; and Mr Mackenzie, whose mind had been turned to
the subject of Hiiidoo science by Lord Napier, appHed for, and
obtained, through Lord Seaforth, a commission in the East
India Company's Engineers on the Madras estabhshment, in
order that he might have a favourable opportunity of prose-
cuting at Madura, the site of the ancient Hindoo college, his
enquiries into the knowledge which the Hindoos possessed, in
early days, of arithmetic, and the diiferent branches of mathe-

h

Iviii INTRODUCTION.

matics. On Mr Mackenzie's arrival at Madras, finding that
my father and mother (the latter being the daughter of his
patron, Lord Napier, and then engaged in completing the life
which had been commenced by her father) were stationed at
Madura, where my father held a poUtical situation of high trust
under his friend Lord Macartney, he obtained leave from Lord
Macartney, the then Governor of Madras, to join them. As
soon as Mr Mackenzie reached Madura, he began his enquiries
relative to the ancient Hindoo college of that place ; and, in
conjunction with my father and mother, formed the plan of
reviving, under the protection of the EngUsh government, the
Hindoo -college. In furtherance of this plan, my father having
obtained from the Nabob of Arcot, the then sovereign of the
country, some deserted ruins in the jungle, about a mile from
the fort of Madura, which were supposed to have been con-
nected in former days with the proceedings of the Hindoo
college, built upon them, at considerable expense, the house
which has ever since been known at that place by the name of
Johnston House, and which is still my property, laying out its
different compartments, under the direction of Mr Mackenzie,
in such a manner as might best suit the adaptation of it as
a building, in which the mathematical instruction that Mr
Mackenzie wished to be circulated amongst all the natives of
the country might be pursued. The pillars which supported

INTRODUCTION. lix

this house were divided into six compartments, upon each of
which all the diagrams were to be carved which were necessary
to illustrate a course of arithmetic, geometry, mechanics,
hydrostatics, optics, and astronomy, there being a building
erected upon the roof, in which plane and spherical trigono-
metry were to be taught : two orreries were to be erected, the
one illustrating the Ptolemaic, the other the Copernican,
system of the universe ; and lectures were to be given in
Tamil, Telugu, Malayalam, and Canarese, pointing out the
superior utihty of the Copernican over the Ptolemaic system,
and the great practical utihty to which the sciences of Europe
might be appUed in every department of practical knowledge.
Mr Mackenzie, shortly affcer he had finished this building for
my father, was obhged to quit Madura on account of the
pubhc service, and the plan of the coUege was, owing to his
absence, not then carried into effect."

It is to be lamented that the hfe of Napier, undertaken by
the nobleman here mentioned, and subsequently by his accom-
pHshed daughter, never saw the hght. In the preface to the
Memoirs pubhshed in 1834, I had erroneously stated that
Mrs Johnston's papers on the subject had perished by fire.
I have since learned, from Sir Alexander himself, that the
papers referred to in his address to the Asiatic Society, and

Ix INTRODUCTION.

which were bound together in the form of a quarto volume,

were lost at sea, on board of the Jane, Duchess of Gordon,

when that vessel perished on her voyage from Ceylon, in 1809.

As Francis, fifth Lord Napier, was he to whom the then

Napier of Culcreuch presented the vahiable manuscripts which

compose the present volume, it is not unUkely that the papers,

from which it appeared that John Napier had prosecuted some

enquiries abroad relative to the history of the Arabic or Indian

notation, had also been obtained from the family of Robert

Napier. The fact that Napier had made this investigation is

the more interesting, that he himself was the first who fully

developed the system. He added to it the Logarithms, and

the reciprocation of the scale in the working of decimal frac-

tions. " There are two improvements," says WaUis (the

friend and contemporary of Newton), " which we have added

to the Algorism of the Arabs, since we received it from them ;

to wit, that of Decimal Fractions and that of Logarithms."

In the manuscripts now printed, and which have so fortu-
nately escaped the fate of nearly the whole of Napier's private
papers iUustrative of his studies, it was not to be expected that
he would aUude to his researches abroad, or enter into anti-
quarian detaUs relative to the origin of the present system of
arithmetical notation. But that he weU understood the prin-

INTRODUCTION. Ixi

ciple of its operation, and had also turned his mind to new
inventions in notation, may be gathered from these manu-
scripts. In the first chapter of the second book of his Arith-
metic, in which he discusses the nomination and notation of
numerical quantities, he says that " every idiom supphes its own
vocal nomination ; but that the written names of integers are
the nine significant figures, — 1, 2, 3, 4, 5, 6, 7, 8, 9 ; and
that these signify various numerical values, according to their
change of place. This progress," he adds, " is fi:*om right to
left ; and the circle 0, which has no signification wherever it be
placed, is merely used to indicate the progress of the significant
figure, by representing the vacancy which that has occasioned
in its progress. When the significant figure is occupying its
first place, it is named," says Napier, " according to its own
individual value ; when it has progressed to the second place, it
is named by its tenfold value ; in the third place, a hundredfold ;
and so on by an infinite progress, each step attaching to the
figure a value equal to the multiphcation of the last step by ten."
Such is Napier's explanation of that aU-powerfiil though simple
expedient ; and none more accurate and lucid has been affbrded
during the two centuries and a half which have elapsed since
he composed his digest of Numbers. The extreme ingenuity
with which he adapted to this principle — of numerical values in
geometrical progression, indicated by the progressive motion

Ijtij INTRODUCTION.

of the same digit in space — a new set of digits of his own
invention, intended for the Arithmetic of Sm*ds, is well worthy
of attention.

Napier eommences what he terms Geometrical Logistic
with an important principle, — his profound consideration and
command of which most probably paved the way to his great
invention. He views quantity and number under two different
aspects or conditions ; namely, either as separated into distinct
parts, capable of being exactly expressed by absolute numbers,
or fractions of numbers, which he terms discrete quantity and
number ; or, as consisting of an infinite continuity of parts,
not to be so expressed by numbers ; and this he terms concrete
quantity or numbers. " If," says he, " 3a'* (by which he means
3 of any given quantity) " refer to three digital hnes, thus,

, it is a discrete number ; but if it refer

to a concrete and continuous tri-digital Une, in this form,

> . , it is called a concrete number; though

not a proper concrete number, but only in relation to the
quantity to which it refers. A proper concrete number,"
Napier adds, " is the root of a number whose root cannot
be measured ; that is, exactly and finitely expressed by any
number, integral or fractional." He had already explained
that the numerical quantity, generally termed the Power of a

INTRODUCTION. Ixiii

number, but which he terms Radicatum, is that which can be
reduced to unit, when divided one or more times by some other
number, the number of the partitions being the index, and the
dividing number the radix, or root. Euler, the great alge-
braical writer of the eighteenth century, had not a more pro-
found knowledge of the properties of the relative numerical
quantities, Root, Power, and Exponent, than Napier in the
sixteenth century had of what he called Radix, Radicatum,
and Index. Indeed, his opening of the subject of involution
is less perplexing than Euler's, whose statement might leave
the student at a loss to know why the square of a number is
called the second power, and not the first. For Euler, after
having said that a power of a number derives its rank, or
particular denominafion, from the number of times it is multi-
pUed by itself, informs us, that a square is obtained by multi-
plying a number once by itself, and a cube, by multiplying a
number twice by itself. Why, then, is a square called the
second power, since the elevation is obtained by the first mul-
tipUcation of the number by itself ? Napier avoids all such
perplexity, when he commences by saying, that the first step in
the process of involution is to " multiply unit by the radix,
which multipHcation returns the radix itself ; secondly, multiply
that again by the radix, and the dupHcatum (the expressive
term he uses for square or second power) is produced ; and so

Ixiv INTRODUCTION.

on, according to the quality of the index, which," he says, " is
determined by the number of .units composing the index."
Thus, whether he explain the nature of a radicated quantity,
as that which a certain number reduces to simple unity by one
or more divisions, or as that which is raised to its particular
denomination by one or more multiplications of the same num-
ber into itself, he so brings unit into his statement of the invo-
lution or evolution, as at once to make manifest the meaning of
the algebraic law», that, although the powers of a number are
the products of the successive multiplications of that number
into itself, yet every number, when thus taken as the root of
successive powers, is considered the first power of itself.

Moreover, Napier did not fail to observe, that there are
certain numbers which neither are to be obtained by the multi-
plication of any number whatever into itself, nor can be resolved
into simple unit by division by any number whatever. Thus,
the number 16 is reduced to 1 by four times dividing by the
number 2. This discovers 2 as what Napier called the quadru-
partient root of 16, and 16 as the quadruplicate, or fourth
power of 2. But there is no number, for instance, which by
involution will produce the number 10. The bipartient or
square root of 9 is 3, because 3 times 3 is 9 ; but what is
the square root of 10 ? In other words, what is the number

INTRODUCTION. Ixv

which, being multipHed by itself, produces 10 ? Not 3 times 3,
that being only 9, nor yet the next number, 4, which gives too
much, namely 16. The doctrine of fractions, indeed, enables
us to express exactly numerical quantities between 3 and 4,
nearer to each other than the relative quantities indicated by
3 and 4, and consequently a nearer approximation to the
quantity sought. This approximation, however, will still be
found to consist of terms, the one too much, and the other too
little, exactly to express what is required ; and a curious pro-
perty it has, that these fractional terms may be brought closer
and closer together, by an endless approximation, and still no
number obtained which, by being multipHed into itself, will
produce the precise number. Hence, numerically speaking,
the number 10 (being a simple example of an infinity of
numbers in the same predicament) has no root ; that is, no
root capable of finite expression in discrete or absolute num-
ber. " Now," says Napier, " a concrete number proper is the
root of such irreduceable number, and these roots are com-
monly called surd and irrational."

Napier had too strong a hold of his subject to reject these
latent and ineffable roots as no quantity at all, or as incapable
of submission to the rules of Arithmetic, and the purposes of
computation. He views them in their proper concrete charac-

Ixvi INTRODUCTION.

ter of quantity or magnitude, rather than as number or multi-
tude ; and calls them " nomina," because susceptible, he says,
rather of being named than numbered. Moreover, he had con-
sidered these quantities so profoundly as to discover all their
computative properties, and fully to illustrate them under the
operation of all the rules of Arithmetic relative to discrete
number and quantity.

Having digested this important chapter of his system of
Logistic, Napier was struck with the necessity of a new nomi-
nation and notation wherewith to work, more easily and effec-
tually, the peculiar quantities in question. The ancient geo-
metricians had derived their nomination of the successive
powers of a number, entirely from the three local dimensions of
nature, — length, breadth, and thickness. Napier, in his Com-
mentaries on the Revelation, took those co-existing quaUties
as an illustration of the Trinity ; but he was not satisfied with
their illustration of the geometrical progression of numbers.
A geometrical progression, of which we hear so much in all
statements of the Logarithmic principle, is so called, because
the successive numerical powers which compose it were named
from the geometrical extensions — represented by a Hne, a
surfaee or square, and a soUd or cube. The square contains
two of the local dimensions, — length and breadth ; in the cube

INTRODUCTION. Ixvii

are the whole three, — ^length, breadth, and thickness. In any
progression of numbers, increasing from unit by a common
ratio of multiplication, the second power contains two dimen-
sions of the root, and was called by the ancient geometricians
a square number, or square ; so the third power, being com-
posed of three dimensions of the root, obtained the name of a
cubic number, or cube. But here the imperfect analogy became
exhausted; for, while the powers of a number are infinite,
extension in space admits of no other distinctions than the
three above mentioned. Hence, by an expedient incongruous
in idea as well as unwieldy in practice, the higher powers were
designated as if repetitions of the geometrical dimensions.
This will be seen from the foUowing table, which is given by
Napier in his earUer work on Algebra, p. 92.

\/-$ Radix quadrata.

^/'TS Radix cubica.

a/"J"J Radix quadrati quadrata.

>s/Jl Radix supersoUda.

\/-5^ Radix quadrati cubica.

's/Jfi Radix secunda supersoUda.

xZ-J-J-^ Radix quadrati quadrati quadrata.

x/TS^TS Radix cubici cubica. Et sic de caBteris in infinitum.

. Here the signs are just an initial abbreviation of the words,
and scarcely less unmanageable in practice. Napier had sub-

Ixviii INTRODUCTION.

sequently become sensible of the defect both of this notatioii
and nomination. The numbers, or rather quantities, particu-
larly dependent upon the nomenclature in question, are surds,
which cannot be expressed in effable numbers, whether in-
tegers or fractions. Accordingly, in his Arithmetic (p. 11),
after showing, for instance, that the surd quantity, called the
cube root of 9, lurks between the numbers 2 and 3, Napier
adds, " But geometricians, studious of greater accuracy, prefer
the mode of prefixing the sign of the index to the radicate
itself, instead of including the root between two terms. For
example, they note the tripartient root of 9 thus, V^C^, which
they pronounce the cube root of 9. I, however, note it thus,
L9, arid call it the tripartient root of 9 ; of which signs I shall
niore fuUy speak in their proper place. Hence arises geome-
trical or concrete numbers, commonly called irrational and
surd." This was tantamount to naming the cube root of 9 (or
whatever power be taken) the third root, and noting it by an
equivalent numeral index ; for we shall immediately see that
the symbol L was an invention of Napier's, to represent the
numeral 3 used for a special purpose,

Along with a nomination expressive of the index, or quality
of the root, it was Napier's object to estabhsh a notation which,
in Hke manner, would immediately suggest to the eye, not

INTRODUCTION. Ixix

merely that a surd root was taken, but what number of root it
was ; and this by symbols entirely new and unappropriated.
Since his day, when roots were to be expressed by radical signs,
instead of by the number itself evolved, the expedient devised
was to retain the contracted R, as the initial of the word root,
and to place within it a small numeral, as the index of the
quahty of the root. Thus ^IO expresses the cube root, or, as
Napier would have called it, the tripartient or third root, of the
number ten considered as a radicatum or power. This nota-
tion, — ^which, it will be observed, gets rid of the unwieldy
expedient of repeating the initials "S, "TS, and fi^ for square,
cube, and supersohd, and presents to the eye the number of
evolutions requisite to obtain the particular root, — was not
invented until many years after Napier's death. Now, the
notation to which he refers, in the sentence quoted above, was
so devised as at once to express that a root, of that number to
which it was attached, was the quantity taken, the particular
number or index of that root, and the fact that the root was a
surd root, or concrete quantity. He took this simple combina-

tion of equal Unes

intersecting each other at equal

intervals. In the nine compartments, which this figure
presents, he inserted the nine numerals, — ^for the sake, he
says, of assisting the memory. From the natural arrange-

Ixx INTRODUCTION.

ment of the figures within, it is easy to remember the

number appertaining to each compartment. Separate the

U12JL5
compartments thus, Jl^EK, and still the memory will retain

TlfelF
the relation of each to the original form, even when the

numerals are withdrawn. By this simple process there is

actually obtained an equivalent for the nine significant digits

of Arithmetic, susceptible of the same combinations, yet per-

fectly distinct in character. Napier's unit of this system is

_I, and when the root which he wishes to represent is beyond

the original value of the highest of his digits, which is F,

corresponding to 9, he obtains the tenth value by moving his

unit bne step to the left, indicating the move by placing a

circle on the right of the significant digit, thus, J**. In like

manner, his unit in its first place combined with the same

advanced a step, thus Jl-i indicates the root of the llth

division, or undecupartient root, as he named it. Thus having

invented a new set of digits equivalent to 1, 2, 3, 4, 5, 6, 7, 8,

9, he submitted them to the same law of progression as in

ordinary arithmetic.

Upon comparing the fragment of Geometrical Logistic
with the first book of his Algebra, which Napier calls the
nominate part of Algebra, it will be obvious, that the latter had

INTRODUCTION. Ixxi

been framed before the conception of the former. Having
composed two books of Algebra, he had extended his plan so
as to embrace the whole art of Logistic ; and the nominate
part of his Algebra would probably have merged in his third
book called Geometrical Logistic, while the second book of
Algebra, which he called the positive or cossic part, in which
unknown quantities are treated of under fictitious representa^
tions, would have formed the fourth and last book of his great
digest of Numbers. It is quite consistent with the nature of
the subject to suppose, that the reason why Napier's book of
the Logistic of concrete quantities by concrete numbers termi-
nates abruptly, and why no other fragment of it could be
found, was, that at this stage of his speculations, he had set
himself to the invention of Logarithms, which may indeed be
considered as the grand result of the Logistic of concrete
quantities by concrete numbers. These, in all probabiHty,
were the very speculations which led him to that achievement,
of which Playfair has said, that, " at a period when the nature
of series, and when every other resource of which he could
avail himself, were so little known, his success argues a depth
and originality of thought, which, I am persuaded, have rarely
been surpassed."

Napier's device for the notation of surds is of itself suffi-

Ixxii INTRODUCTION.

cient to prove that his invention of the Logarithms was sub-
sequent to such speculations ; for the mind which accomplished
the Logarithms, and compounded that name for them, could
scarcely fail very soon to perceive that the radical signs, expres-
sive of a surd root, were to be entirely superseded by fractional
exponents. Dr Hutton committed a strange anachronism, and
one which does injustice to the genius of Napier, in the follow-
ing passage : — " The notation of powers and roots, by the pre-
sent mode of exponents, has introduced a new and general
arithmetic of exponents or powers ; for hence powers are
multipHed by only adding their exponents, divided by subtract-
ing the exponents, raised to other powers, or roots of them
extraC»ted, by multiplying or dividing the exponent by the
index of the power or root. So a'^y.d^-=.a^ ^ and a\Y.ah-=.a\ \
rt^-f-a5=fl2, and a\-^\-=.a\ ; the second power of «' is a^^ and
the third root of «^ is o^, This algorithm of powers led
the way to the invention of Logarithms, which are only the
indices or exponents of powers ; and hence the addition and
subtraction of Logarithms answer to the multipHcation and
division of numbers, while the raising of powers and the ex-
tracting of roots is eflfected by multiplying the logarithm by the
index of the power, or dividing the logarithm by the index of
the root." — Math. Dict. If the algorithm of which Dr Hutton
here speaks had belonged to science before Napier invented

INTRODUCTION. Ixxiii

the Logarithms, Playfair's eulogy of him, quoted above, would
not have been merited ; for the arithmetic of powers and
exponents might have disclosed the Logarithms even to an
ordinary mathematician. The more accurate statement is,
that the invention of Logarithms led the way to the algorithm
of powers. Napier, — writing, be it remembered, some time in
the sixteenth century, before Vieta, Harriot, Girard, and
Oughtred, and when Algebra was not cultivated at all in this
country, — ^had no assistance from the algebraic refinement of
working known quantities by means of other symbols than the
significant digits, or of expressing powers by means of small
letters instead of numerals. He did not, for instance, consider
aaaa as the quadrupUcatum (to use his own term) of any
number a ; still less did he consider the same quantity in this
form, a^ ; he had neither the Hteral notation of powers, nor the
numeral notation of indices ; for although, in explaining their
genesis, he named the indices, one, two, three, &c., and also
noted and arranged them above the powers by the numerals 1,
2, 3, &c., yet he did not systematically attach them to the root
for the expression of the power. To have done so would have
been to anticipate the notation of Descartes, whose epoch is
1637, twenty years after Napier's death. But, while he had
not these advantages to lead him to the Logarithms, the
arithmetic of exponents was so obviously to be deduced from

k

Ixxiv ^ INTRODUCTION.

that invention, that, having achieved the Logarithms, he could
not fail immediately to perceive the application of fractional
indices for the expression of roots, — and, consequently, that his
proposed notation of surds, in his Geometrical Logistic, was
comparatively crude and useless. This fact, which it would
occupy too much space fiilly to illustrate here, may be oifered
as internal evidence, affbrded by the manuscripts now printed,
that they are studies which must have preceded the conception
of his great invention.

That which may truly be said to have led Napier to the
invention of Logarithms, is the profound view he took of
the Logistic of concrete quantity, after having thoroughly mas-
tered that of discrete quantity. The properties of what we
may now term the Logarithms of discrete ratios, namely,
the series of natural numbers taken to enumerate the steps or
terms of any geometrical progression represented by integers,
had been known since the days of Archimedes. But the
desideratum was, to prove that the infinite series of natural
numbers, 1, 2, 3, 4, 5, 6, &c., might be viewed as in a geome-
trical progression, and then to discover the indices of all these
numbers, considered as powers of some given number, — which
would disclose the corresponding arithmetical series, and the
Logarithms par excellence. Less profoundly considered, the

INTRODUCTION. Ixxv

idea that the arithmetical series, or progression by equi-
difference of the natural numbers, could be viewed as a
geometrical series, or progression by multipHcation, seemed
a contradiction in terms. Napier commences by demon-
strating that the progressive increase, or decrease, of a con-
crete or continuous quantity may be conceived to be gener-
ated by a motion in space, so regulated, in respect of the
velocity, as to generate a continuous geometrical progres-
sion, of infinitely small ratios of magnitude, various terms of
which might be represented, or infinitely nearly so, by the
series of natural numbers, which thus all become terms of
a geometrical progression. In Uke manner, he demonstrated
the genesis of a corresponding arithmetical progression by a
simultaneous motion, the velocity of which, being equal through-
out and not increasing or decreasing, generated magnitudes
in an arithmetical progression, which, at any given point of the
progress, might be represented by a numerical expression that
would serve for the index, or number of the ratios (logarithm),
of the corresponding point in the simultaneous geometrical
motion. His idea of motion, thus taken to generate propor-
tional magnitudes, was analogous to the law of the Arabic
notation in discrete numbers, where the significant digit may
be conceived to generate a decuple progression, by traveUing
in a Une from right to left. There are various circumstances

Ixxvi INTRODUCTION.

connected with the doctrine of Logarithms, unnecessary to be
considered here, showing its natural affinity to the system of the
Arabic notation, which in fact only became perfectly developed
when the Logarithms were invented ; and it is singular that,
among the mathematical stores of those distant climes from
which was derived the refined and powerful notation in question,
and tliroughout the long ages of its operation in the hands of
genius, not a trace can be discovered of a conception of that
development of the scale wliich Napier accompUshed in the
Logarithms and the arithmetic of Decimal fractions. The
Chinese are said to have laid claim to the invention ; but the
splendid copy of the Logarithms which issued from the imperial
press (5f Pekin, contains cei^tain errors which have been recently
discovered in the European tables, previously pubHshed.

Napier's mode of demonstrating the Logarithms, by the
motion of points (fluxu puncti) generating two lines — the one
" when the point describing the same goeth forward equal
spaces in equal times or moments," and the other (which,
to facilitate his operations, he took in the decreasing ratio,
having complete command of the arithmetic of negative
quantities,) " when the point, describing the same in equal
times, cutteth off parts continually of the same proportion
to the lines from which they are cut ofi*," — was afterwards

INTRODUCTION. Ixxvii

adopted by Sir Isaac Newton, in similar terms, to illustrate
his own discovery of Fluxions. The accidental circumstance,
or first idea, which may have led any great inventor into
the path of his distinction, can rarely be discovered. I have
endeavoured to illustrate Napier's progress to the Loga-
rithms through his mathematical studies now pubHshed.
Newton himself, in his treatise of the Quadrature of Curves,
announces the method that led him to his great discovery of
Fluxions. " I consider," he says, " mathematical quantities
in this place not as consisting of very small parts, but as
described by a continued motion. Lines are described, and
therefore generated, not by the opposition of parts, but by the
continued motion of points ; superficies by the motion of hnes ;
soUds by the motion of superficies ; angles by the rotation of
the sides ; portions of time by a continual flux ; and so in
other quantities. These geneses really take place in the nature
of things, and are daily seen in the motion of bodies. And
after this manner the ancients, by drawing moveable right
hnes along immoveable right Hnes, taught the genesis of rect-
angles. Therefore, considering that quantities which increase
in equal times, and by increasing are generated, become greater
or less according to the greater or less velocity with which they
increase and are generated, I sought a method of determining
quantities fi^om the velocities of the motion or increments

]xxviii INTRODUCTION.

with which they are generated ; and calling these velocities of
the motions or increments Fluxions, and the generated quanti-
ties Fluents, I fell by degrees upon the method of Fluxions,
which I have made use of here in the quadrature of curves, in
the years 1665 and 1666." It is remarkable, that, instead of
merely referring to the ancients in this passage, Newton had
not rather said, — ' And after this manner Napier, by drawing
a moveable point along a right line, taught the genesis of
Logarithms ; and when I speak of quantities becoming greater
or less according to the greater or less velocity with which the
increase and decrease are generated, and of determining quan-
tities from the velocities of the motions or increments with
which they are generated, — and when I call these velocities
of the motions or increments Fluxions, — I avail myself of
Napier's demonstration, I adopt his language, — and even his
very expressions, " fluxu," and " incrementi aut decrementi."'
Newton may have closely studied Napier's published works, or
he may never have seen them. The first idea is suggested by
the above remarkable coincidences of thought and expression ;
and also by the fact that Newton's Commentaries On Scriptural
prophecy is little else than a less elaborate repetition of Napier's.
The other idea, however, is after all the more Hkely, from this
circumstance, that, throughout the voluminous published works
of Newton, mathematical and theological, no notice of Napier

INTRODUCTION. Ixxix

is to be found. Yet the great lever with which Newton worked
was the Logarithms ; and the Binomial Theorem, says Baron
Maseres, " is so very closely connected with the subject of
Logarithms, as to be the foundation of the best methods of
computing them."

In Napier's Arithmetic, (p. 50,) which Newton certainly
never saw, there is a figurate diagram, worthy of especial
notice, as being an unknown anticipation of the Arithmetical
Triangle of the celebrated Blaise Pascal ; presented, however,
in a far more beautiful form than that of the French mathe-
matician. The properties of this triangle are so intimately
connected with the Binomial Theorem, that Bernoulh, on that
account, somewhat rashly claims for Pascal the merit of the
invention. " Nous avons trouv^," he says, " ce merveilleux
theoreme aussi bien que M. Newton, d'une maniere plus simple
que la sienne. Feu M. Pascal a ete le premier qui Pa
inventee." Maseres, who repubhshed PascaPs mathematical
works, says of them, — " These works are so fuU of genius and
invention, that I thought I should do a service to the mathe-
maticians of Great Britain by repubhshing them in this collec-
tion. Some of them, and more especiahy his Arithmetical
Triangle, have a considerable connexion with Logarithms,
by aflfording a good demonstration of Sir Isaac Newton's

I^yy INTRODUCTION.

Binomial Theorem, in the case of integral and affirmative
powers, which is of great use in the construction of Loga^
rithms." But Napier had anticipated Pascal in this invention,
which, however, only received its most valuable algebraic
appHcation, dependent as that is upon the modern exponential
notation, at the hands of Sir Isaac Newton.

There are other indications, in the manuscripts now sub-
mitted to the public, of an inventive genius in Numbers, far
before the times in which Napier wrote. Professor Playfair,
in his Dissertation on the Progress of Mathematical and
Physical Science, when speaking of Albert Girard, — a Flemish
matliematician, whose principal work, Invention Nouvelle en
Algebre, was printed in 1669, — observes, " He appears to
have been the first who understood the use of negative roots in
the solution of geometrical problems, and is the author of the
figurative expression which gives to negative quantities the name
of ' quantities less than nothing,' — a phrase that has been severe-
ly censured by those who forget that there are correct ideas
which correct language can hardly be made to express. The
same mathematician conceived the notion of imaginary roots,
and showed that the number of the roots of an equation could
not exceed the exponent of the highest power of the unknown
quantity." Sir John Leslie, who continued Playfair's Disser-

INTRODUCTION. Ixxxi

tation, also comments upon Girard's introduction of the phra^
seology, ' quantities greater and less than nothing,' and upon
his discovery and nomenclature of impossible quantities. It
is remarkable that Playfair and Leshe, who have expressed
great admiration for the genius of Napier, had not looked at
or studied his pubHshed works. Napier's command of the
arithmetic of -{• and — , the signs of positive and negative
quantities, (or as he more properly phrased it, abundant or
abounding, and defective quantities,) was of great assistance
to him in reahsing his conception of Logarithms ; and, in the
first chapter of his Canon Mirificus, he says, (to quote from
the EngUsh translation of 1616, revised by himself,) " There-
fore we call the logarithms of the sines abounding, because
they are always greater than nothing, and set this mark -f- before
them, or else none ; but the logarithms which are less than
nothing we call defective or wanting, setting this mark —
before them." Dr Horsley had fallen into the same mistake of
attributing the origin of this phraseology to Girard. But as
for the quantity called impossible, these authors had not the
means of knowing that Napier was the person who first disco^
vered the use of it in mathematics, and that he had somewhat
exultingly recorded the fact. Having laid the foundation, (p. 19,)
by an exposition of the arithmetic of + and — not inferior to
Euler's, he makes the announcement, in his Logistica Geome-

Ixxxil INTRODUCTION.

trica, which I shall here translate. " Seeing, therefore, that a
8urd uninome may be the root either of an abounding or of a
defective number, and that its index may be either even or odd,
from this fourfold cause it foUows, that some surds are abound-
ing, some defective, some both abounding and defective, which I
term ' gemina ;' some neither abounding nor defective, which I
call ' nugacia.' The foundation of this gi-eat Algebraic secret
I have already laid in the sixth chapter of the first book ; and,
though hitherto unrevealed by any one else, so far as I know,
the value of it to this art, and to Mathematics in general, shall
presently appear."

There can be no doubf that by " nugacia" Napier means
the impossible quantity, and that he was the very first to con-
ceive the idea, and to propose its use, in the arithmetic of
surds and the theory of equations. He explains precisely its
nature, and gives rules for its notation, and applies to it the
startling designations of " a quantity absurd and impossible,
nonsensical and signifying nothing." He shows that all roots,
whether abounding or defective, (or, as he also terms it,
greater or less than nothing,) when multipUed to an even index
produce an abounding radicate. Thus, take for the root
H-2, and let it be multipHed to the index 4, (in other words,
raised to the fourth power,) and the radicate will be +16 ; in

INTRODUCTION. Ixxxiii

like manner, raise — 2 to the index 4, and the radicate will be
+16; so — 2 and +2 are equally the quadripartient root of
16. Hence, says he, every abounding radicate, having an even
index, has two roots. These were what he called geminal.
But, he adds, if roots both abounding and defective belong
to the abounding radicate with an even index, then there is no
root of any kind left for the same radicates having the defective
sign prefixed. Hence, for example, Z\ — 16 (in modern nota^
tion, ^-16 ) is an impossible quantity. The " great emolu-
ment" which Napier expected to bestow upon Mathematics by
this ghost of a quantity, can only be well understood by pro-
found mathematicians. Euler has devoted a chapter to impos-
sible or imaginary quantities, which he concludes with these
observations : " It remains for us to remove any doubt which
may be entertained concerning the utUity of the numbers of
which we have been speaking ; for those numbers being impos-
sible, it would not be surprising if they were thought entirely
useless, and the object only of an unfounded speculation. This,
however, would be a mistake ; for the calculation of imaginary
quantities is of the greatest importance, as questions frequently
arise of which we cannot immediately say whether they include
any thing real and possible or not ; but when the solution of
such a question leads to imaginary numbers, we are certain
that what is required is impossible."

Ixxxiv INTRODUCTION.

But it appears, that, when Napier had broken ground
upon the great field of equations, and with a vigour and com-
mand of the subject which obviously was about to anticipate
the conquests of many of his illustrious successors, he was
arrested in that progress by another idea. The stars were
becoming too many for Tycho and Kepler ; so he promised
them the Logarithms. His mind had already penetrated the
Arabic system in every direction. He had already orderly set
down how the various operations of number and quantity gra-
dually unfold, — ^how the vast fabric produces itself, growth after
growth, every rule the parent of another, and the whole inti-
mately related, in all its parts, in endless generations of Num-
bers. Having traced the genealogy of Numbers upwards from
nothing, he had taken zero as the pivot for a reciprocal scale,
and unfolded the logistic of " quantitates minores nihilo," and
'•' quantitates impossibiles et nihil significantes." Then he had
viewed unit as broken into another infinite scale, and had
reduced to order all the operations of Arithmetic upon its
fractions. It remained to condense the vast system into greater
simpHcity and power, and also to display unit as the pivot for a
reciprocal play of the Arabic scale, as he had done by zero. This
he accompHshed by his invention and calculation of Logarithms.
Moreover, in his brilHant performance of that promise to
Tycho, he first appHed — before either Newton or Leibnitz —

INTRODUCTION. Ixxxv

the great doctrine of Variable Quantity, and thus he not merely
afforded materials for, but, to this extent, had actually antici-
pated, the algebraic conquests of Newton. Newton is well de-
signed the Prince of Mathematicians. Napier is the King of
Numbers.

It adds not a Httle to the interest with which these manu-
scripts must be viewed, that they had been written out for
Henry Briggs, the sateUite of Napier, and to whom Napier
himself refers as " amico mihi longe carissimo." Briggs was
ten years younger than Napier, and, about the year 1596,
had been appointed Professor of Geometry, in the munificent
estabHshment founded by Sir Thomas Gresham, where he de-
voted himself particularly to Astronomy, and became known
to the most celebrated men of his day. In a letter to Arch-
bishop Usher dated August 1610, Briggs says, " Concerning
eclypses, you see by your own experience that good purposes
may in two years be honestly crossed ; and therefore, till you
send me your tractate you promised the last year, do not look
for much from me ; for if any other business may excuse, it will
serve me too. Yet I am not idle in that kind, for Kepler hath
troubled all, and erected a new frame for the motions of all the
seven upon a new foundation, making scarce any use of any
former hypotheses ; yet I dare not much blame him, save that

Ixxxvi INTRODUCTION.

he is tedious and obscure, and at length coming to the point,
he hath left out the principal verb — I mean his tables both of
middle motion and prosthaphsereseon — reserving all, as it seem-
eth, to his Tab. Rudolpheus, setting down only a lame pattem
in Mars ; but I think I shall scarce with patience expect his
next books, unless he speed himself quickly." But ere those
long promised Tables were published, the Logarithms appeared,
and Kepler immediately remodeled his work upon this new
chapter in science. Briggs, however, was the first to catch fire
at the discovery. In another letter to Usher, dated Gresham
House, lOth March 1615, after speaking of the Arabic versions
of the Greek philosophers, he adds, " Napper, Lord of Markin-
ston, hath set my head and hands arwork with his new and
admirable Logarithms : I hope to see him this summer, if it
please God ; for I never saw book which pleased me better, or
made me more wonder : I purpose to discourse with him con-
cerning ecHpses, for what is there which we may not hope for at
his hands ? " Dr Thomas Smith, the biographer of Usher and
Briggs, has painted in vivid colours the state of excitement into
which the latter was thrown by the Canon Mirificus. He says
that Ursin, Kepler, Frobenius, Batschius, and others, received
it with great honour, but none more so than Briggs. " He
cherished it as the apple of his eye ; it was ever in his bosom,
or his hand, or pressed to his heart ; with greedy eyes, and

INTRODUCTION. Ixxxvii

mind absorbed, he perused it again and again. In his study, or
in his bed, his whole thoughts were bent upon illustrating it,
and bringing it to the last stage of perfection ; he considered
that his study could not be more fruitfully, or beautifully, or
gloriously bestowed than upon this most illustrious discipHne ;
for he regarded all other work as idleness ; it was the theme of
his praise in familiar conversation with his friends, and ex
cathedra he expounded it to his disciples."

Mr Hallam, in his Introduction to the Literature of Europe,
recently pubhshed, has not failed to give a prominent place to
Napier. But it is to be regretted that he had not derived his
information, relative to Napier's works, from accurate sources ;
and I may take this opportunity of correcting some mistakes
into which he has fallen. Speaking of the Logarithms, he says,
" This Napier first pubHshed in 1614, with the title Logarith-
morum Canonis Descriptio, seu Arithmeticarum Supputatio-
num Mirabihs Abbreviatio. He died in 1618 ; and in a post-
humous edition, entitled Mirifici Logarithmorum Canonis De-
scriptio, 1618, the method of construction, which had been at
first withheld, is given ; and the system itself, in consequence
perhaps of the suggestion of his friend Briggs, underwent some
change." But the real title of Napier's original pubHcation is,
" Mirifici Logarithmorum Canonis Descriptio ; ejusque usus in

Ixxxviii INTRODUCTION.

utrdque Trigonometrifi, ut etiam in omni Logisticd Mathe-
matic^, amplissimi, faeillimi, et expeditissimi Explicatio." It is
of importance to preserve the true title, because it bears evi-
dence on the face of it, that, although (probably in reference
to Tycho) he specially adapted his system to Trigonometry,
he was perfectly aware of its value to the whole analytic art —
" in omni Logistica Mathematicd." Napier died in 1617. Nor
am I aware of any such posthumous edition of the Canonis
Descriptio, dated in 1618. Probably this is an inaccurate
reference to the distinct work of Napier's, abeady mentioned,
pubHshed in 1619, under the joint editorship of Robert Napier
and Henry Briggs, and along with which appeared a reprint
of thfe former work. The title of the Constructio, the most
beautiftd of Napier's works, is, — " Mirifici Logarithmorum
Canonis Constructio ; et eorum ad naturales ipsorum numeros
Habitudines ; una cum Appendice de alia e^que praestantiore
Logarithmorum specie condenda. Quibus accessere Proposi-
tiones ad triangula spherica facihore calculo resolvenda. Una
cum Annotationibus aliquot doctissimi D. Henrici Briggii in
eas et memoratam appendicem." In the Memoirs of Napier
I had called attention to the fact, that his works were more
frequently referred to than examined ; and that even Playfair,
Leshe, and Dr Horsley were not acquainted with the original
institute of the Logarithms. M. Biot, accordingly, when

INTRODUCTION. Ixxxix

reviewing the Memoirs, had sought out the original editions
of Napier's works, and he appears to have been much struck
with their power. Referring to the development, and loga-
rithmic appUcation, of the algebraic calculus since Napier's
time, he adds, — " I declare, however, to the honour of Napier,
that these means produce nothing which cannot be very easily
attained by his own method ; and if, as it is natural to sup-
pose, this assertion may seem to our analysts something more
than rash, I hope presently to aflPord unanswerable proofs of
its accuracy. But to form this just idea of Napier's operations,
it is necessary to study his own works, especially his second
work, in which he unfolds his method, — and not to rely upon
extracts." In reference to the above, M. Biot prefixes to his
review an investigation which he entitles, " Analyse et re- *
stitution de 1'ouvrage original de Napier, intitule, Mirifici
Logarithmorum Canonis Constructio."

In the title-page of the Constructio, of which Briggs him-
self was one of the editors, will be found an allusion to Napier's
improvement of his own system ; and in the preface, the im-
provement is declared to have originated with Napier himself,
nor is there any allusion to a suggestion from Briggs. In his
letter to the Chancellor, Napier also mentions the improvement
he intended, without any reference to Briggs. Indeed, the

m

xc INTRODUCTION.

original edition of his Canon contains an allusion to the same,
before Briggs knew of the Logarithms. Mr Hallam's doubt
on the subject, which he thus repeats in another passage, —
" It is uncertain from which of them the change in the form
of Logarithms proceeded," — would not have occurred had he
examined these works. We have the assurance both of Napier
and Briggs that there is no dubiety about the matter. Indeed,
Briggs has recorded the exact state of the case in an interest-
ing statement prefixed to his Arithmetica Logarithmica, pub-
lished in London in 1624, and which I shall here translate : —
" That these Logarithms differ from those which that illus-
trious man, the Baron of Merchiston, pubUshed in his Canon
Mirificus, must not surprise you. For I myself, when ex-
pounding publicly in London their doctrine to my auditors in
Gresham CoUege, remarked that it would be much more con-
venient that should stand for the logarithm of the whole
sine, as in the Canon Mirificus ; but that the logarithm of the
tenth part of the same whole sine, that is to say, 5 degrees,
44 minutes, and 21 seconds, should be 10,000,000,000. Con-
cerning that matter I wrote immediately to the author himself ;
and, as soon as the season of the year, and the vacation-time
of my public duties of instruction permitted, I took a journey to
Edinburgh, where, being most hospitably received by Napier,
I stuck to him for a whole month. But, as we held discourse

INTRODUCTION. xci

concerning this change in the system of Logarithms, he said
that for a long time he had been sensible of the same thing,
and had been anxious to accomplish it, but had pubhshed what
he had already prepared, until he could construct tables more
convenient, if other weighty matters, and the frail state of his
health would suffer him so to do. But, he coriceived, the
change ought to be effected in this manner, that should
become the logarithm of unity, and 10,000,000,000 that of the
whole sine. This I could not but admit was by far the fittest
modification ; so, casting aside what I had already prepared,
I commenced under his advice to bend my mind to the calcu-
lation of these tables ; and in the foUowing summer I again
took journey to Edinburgh, where I submitted to him the
principal part of those tables which are here pubhshed ; and I
was about to do the same even the third summer, had it pleased
God to have spared him to us so long."

It is not for the sake of asserting his right to what, at all
events, is a simple derivative idea, that these proofs are referred
to. It is because the dubiety expressed by Mr Hallam might
infer that Napier had done injustice to Briggs, whose genius he
appreciated, and whose friendship and admiration he cordially
returned. " I will acquaint you," says Lilly, in his Life and
Time