Napier's work (which will henceforth in this article be referred to as
the _Descriptio_) immediately on its appearance in 1614 attracted the
attention of perhaps the two most eminent English mathematicians then
living--Edward Wright and Henry Briggs. The former translated the work
into English; the latter was concerned with Napier in the change of the
logarithms from those originally invented to decimal or common
logarithms, and it is to him that the original calculation of the
logarithmic tables now in use is mainly due. Both Napier and Wright died
soon after the publication of the _Descriptio_, the date of Wright's
death being 1615 and that of Napier 1617, but Briggs lived until 1631.
Edward Wright, who was a fellow of Caius College, Cambridge, occupies a
conspicuous place in the history of navigation. In 1599 he published
_Certaine errors in Navigation detected and corrected_, and he was the
author of other works; to him also is chiefly due the invention of the
method known as Mercator's sailing. He at once saw the value of
logarithms as an aid to navigation, and lost no time in preparing a
translation, which he submitted to Napier himself. The preface to
Wright's edition consists of a translation of the preface to the
_Descriptio_, together with the addition of the following sentences
written by Napier himself: "But now some of our countreymen in this
Island well affected to these studies, and the more publique good,
procured a most learned Mathematician to translate the same into our
vulgar English tongue, who after he had finished it, sent the Coppy of
it to me, to bee seene and considered on by myselfe. I having most
willingly and gladly done the same, finde it to bee most exact and
precisely conformable to my minde and the originall. Therefore it may
please you who are inclined to these studies, to receive it from me and
the Translator, with as much good will as we recommend it unto you."
There is a short "preface to the reader" by Briggs, and a description of
a triangular diagram invented by Wright for finding the proportional
parts. The table is printed to one figure less than in the _Descriptio_.
Edward Wright died, as has been mentioned, in 1615, and his son, Samuel
Wright, in the preface states that his father "gave much commendation of
this work (and often in my hearing) as of very great use to mariners";
and with respect to the translation he says that "shortly after he had
it returned out of Scotland, it pleased God to call him away afore he
could publish it." The translation was published in 1616. It was also
reissued with a new title-page in 1618.
Henry Briggs, then professor of geometry at Gresham College, London, and
afterwards Savilian professor of geometry at Oxford, welcomed the
_Descriptio_ with enthusiasm. In a letter to Archbishop Usher, dated
Gresham House, March 10, 1615, he wrote, "Napper, lord of Markinston,
hath set my head and hands a work with his new and admirable logarithms.
I hope to see him this summer, if it please God, for I never saw book
which pleased me better, or made me more wonder.[1] I purpose to
discourse with him concerning eclipses, for what is there which we may
not hope for at his hands," and he also states "that he was wholly taken
up and employed about the noble invention of logarithms lately
discovered." Briggs accordingly visited Napier in 1615, and stayed with
him a whole month.[2] He brought with him some calculations he had
made, and suggested to Napier the advantages that would result from the
choice of 10 as a base, an improvement which he had explained in his
lectures at Gresham College, and on which he had written to Napier.
Napier said that he had already thought of the change, and pointed out a
further improvement, viz., that the characteristics of numbers greater
than unity should be positive and not negative, as suggested by Briggs.
In 1616 Briggs again visited Napier and showed him the work he had
accomplished, and, he says, he would gladly have paid him a third visit
in 1617 had Napier's life been spared.
Briggs's _Logarithmorum chilias prima_, which contains the first
published table of decimal or common logarithms, is only a small octavo
tract of sixteen pages, and gives the logarithms of numbers from unity
to 1000 to 14 places of decimals. It was published, probably privately,
in 1617, after Napier's death,[3] and there is no author's name, place
or date. The date of publication is, however, fixed as 1617 by a letter
from Sir Henry Bourchier to Usher, dated December 6, 1617, containing
the passage--"Our kind friend, Mr Briggs, hath lately published a
supplement to the most excellent tables of logarithms, which I presume
he has sent to you." Briggs's tract of 1617 is extremely rare, and has
generally been ignored or incorrectly described. Hutton erroneously
states that it contains the logarithms to 8 places, and his account has
been followed by most writers. There is a copy in the British Museum.
Briggs continued to labour assiduously at the calculation of logarithms,
and in 1624 published his _Arithmetica logarithmica_, a folio work
containing the logarithms of the numbers from l to 20,000, and from
90,000 to 100,000 (and in some copies to 101,000) to 14 places of
decimals. The table occupies 300 pages, and there is an introduction of
88 pages relating to the mode of calculation, and the applications of
logarithms.
There was thus left a gap between 20,000 and 90,000, which was filled up
by Adrian Vlacq (or Ulaccus), who published at Gouda, in Holland, in
1628, a table containing the logarithms of the numbers from unity to
100,000 to 10 places of decimals. Having calculated 70,000 logarithms
and copied only 30,000, Vlacq would have been quite entitled to have
called his a new work. He designates it, however, only a second edition
of Briggs's _Arithmetica logarithmica_, the title running _Arithmetica
logarithmica sive Logarithmorum Chiliades centum, ... editio secunda
aucta per Adrianum Vlacq, Goudanum_. This table of Vlacq's was
published, with an English explanation prefixed, at London in 1631 under
the title _Logarithmicall Arithmetike ... London, printed by George
Miller_, 1631. There are also copies with the title-page and
introduction in French and in Dutch (Gouda, 1628).
Briggs had himself been engaged in filling up the gap, and in a letter
to John Pell, written after the publication of Vlacq's work, and dated
October 25, 1628, he says:--
"My desire was to have those chiliades that are wantinge betwixt 20
and 90 calculated and printed, and I had done them all almost by my
selfe, and by some frendes whom my rules had sufficiently informed,
and by agreement the busines was conveniently parted amongst us; but I
am eased of that charge and care by one Adrian Vlacque, an Hollander,
who hathe done all the whole hundred chiliades and printed them in
Latin, Dutche and Frenche, 1000 bookes in these 3 languages, and hathe
sould them almost all. But he hathe cutt off 4 of my figures
throughout; and hathe left out my dedication, and to the reader, and
two chapters the 12 and 13, in the rest he hath not varied from me at
all."
The original calculation of the logarithms of numbers from unity to
101,000 was thus performed by Briggs and Vlacq between 1615 and 1628.
Vlacq's table is that from which all the hundreds of tables of
logarithms that have subsequently appeared have been derived. It
contains of course many errors, which were gradually discovered and
corrected in the course of the next two hundred and fifty years.
The first calculation or publication of Briggian or common logarithms of
trigonometrical functions was made in 1620 by Edmund Gunter, who was
Briggs's colleague as professor of astronomy in Gresham College. The
title of Gunter's book, which is very scarce, is _Canon triangulorum_,
and it contains logarithmic sines and tangents for every minute of the
quadrant to 7 places of decimals.
The next publication was due to Vlacq, who appended to his logarithms of
numbers in the _Arithmetica logarithmica_ of 1628 a table giving log
sines, tangents and secants for every minute of the quadrant to 10
places; these were obtained by calculating the logarithms of the natural
sines, &c. given in the _Thesaurus mathematicus_ of Pitiscus (1613).
During the last years of his life Briggs devoted himself to the
calculation of logarithmic sines, &c. and at the time of his death in
1631 he had all but completed a logarithmic canon to every hundredth of
a degree. This work was published by Vlacq at his own expense at Gouda
in 1633, under the title _Trigonometria Britannica_. It contains log
sines (to 14 places) and tangents (to 10 places), besides natural sines,
tangents and secants, at intervals of a hundredth of a degree. In the
same year Vlacq published at Gouda his _Trigonometria artificialis_,
giving log sines and tangents to every 10 seconds of the quadrant to 10
places. This work also contains the logarithms of numbers from unity to
20,000 taken from the _Arithmetica logarithmica_ of 1628. Briggs
appreciated clearly the advantages of a centesimal division of the
quadrant, and by dividing the degree into hundredth parts instead of
into minutes, made a step towards a reformation in this respect, and but
for the appearance of Vlacq's work the decimal division of the degree
might have become recognized, as is now the case with the corresponding
division of the second. The calculation of the logarithms not only of
numbers but also of the trigonometrical functions is therefore due to
Briggs and Vlacq; and the results contained in their four fundamental
works--_Arithmetica logarithmica_ (Briggs), 1624; _Arithmetica
logarithmica_ (Vlacq), 1628; _Trigonometria Britannica_ (Briggs), 1633;
_Trigonometria artificialis_ (Vlacq), 1633--have not been superseded by
any subsequent calculations.
In the preceding paragraphs an account has been given of the actual
announcement of the invention of logarithms and of the calculation of
the tables. It now remains to refer in more detail to the invention
itself and to examine the claims of Napier and Briggs to the capital
improvement involved in the change from Napier's original logarithms to
logarithms to the base 10.
The _Descriptio_ contained only an explanation of the use of the
logarithms without any account of the manner in which the canon was
constructed. In an "Admonitio" on the seventh page Napier states that,
although in that place the mode of construction should be explained, he
proceeds at once to the use of the logarithms, "ut praelibatis prius
usu, et rei utilitate, caetera aut magis placeant posthac edenda, aut
minus saltem displiceant silentio sepulta." He awaits therefore the
judgment and censure of the learned "priusquam caetera in lucem temerè
prolata lividorum detrectationi exponantur"; and in an "Admonitio" on
the last page of the book he states that he will publish the mode of
construction of the canon "si huius inventi usum eruditis gratum fore
intellexero." Napier, however, did not live to keep this promise. In
1617 he published a small work entitled _Rabdologia_ relating to
mechanical methods of performing multiplications and divisions, and in
the same year he died.
The proposed work was published in 1619 by Robert Napier, his second son
by his second marriage, under the title _Mirifici logarithmorum canonis
constructio_.... It consists of two pages of preface followed by
sixty-seven pages of text. In the preface Robert Napier says that he has
been assured from undoubted authority that the new invention is much
thought of by the ablest mathematicians, and that nothing would delight
them more than the publication of the mode of construction of the canon.
He therefore issues the work to satisfy their desires, although, he
states, it is manifest that it would have seen the light in a far more
perfect state if his father could have put the finishing touches to it;
and he mentions that, in the opinion of the best judges, his father
possessed, among other most excellent gifts, in the highest degree the
power of explaining the most difficult matters by a certain and easy
method in the fewest possible words.
It is important to notice that in the _Constructio_ logarithms are
called artificial numbers; and Robert Napier states that the work was
composed several years (_aliquot annos_) before Napier had invented the
name logarithm. The _Constructio_ therefore may have been written a good
many years previous to the publication of the _Descriptio_ in 1614.
Passing now to the invention of common or decimal logarithms, that is,
to the transition from the logarithms originally invented by Napier to
logarithms to the base 10, the first allusion to a change of system
occurs in the "Admonitio" on the last page of the _Descriptio_ (1614),
the concluding paragraph of which is "Verùm si huius inventi usum
eruditis gratum fore intellexero, dabo fortasse brevi (Deo aspirante)
rationem ac methodum aut hunc canonem emendandi, aut emendatiorem de
novo condendi, ut ita plurium Logistarum diligentia, limatior tandem et
accuratior, quàm unius opera fieri potuit, in lucem prodeat. Nihil in
ortu perfectum." In some copies, however, this "Admonitio" is absent. In
Wright's translation of 1616 Napier has added the sentence--"But because
the addition and subtraction of these former numbers may seeme somewhat
painfull, I intend (if it shall please God) in a second Edition, to set
out such Logarithmes as shall make those numbers above written to fall
upon decimal numbers, such as 100,000,000, 200,000,000, 300,000,000,
&c., which are easie to be added or abated to or from any other number"
(p. 19); and in the dedication of the _Rabdologia_ (1617) he wrote
"Quorum quidem Logarithmorum speciem aliam multò praestantiorem nunc
etiam invenimus, & creandi methodum, unà cum eorum usu (si Deus
longiorem vitae & valetudinis usuram concesserit) evulgare statuimus;
ipsam autem novi canonis supputationem, ob infirmam corporis nostri
valetudinem, viris in hoc studii genere versatis relinquimus: imprimis
verò doctissimo viro D. Henrico Briggio Londini publico Geometriae
Professori, et amico mihi longè charissimo."
Briggs in the short preface to his _Logarithmorum chilias_ (1617) states
that the reason why his logarithms are different from those introduced
by Napier "sperandum, ejus librum posthumum, abunde nobis propediem
satisfacturum." The "liber posthumus" was the _Constructio_ (1619), in
the preface to which Robert Napier states that he has added an appendix
relating to another and more excellent species of logarithms, referred
to by the inventor himself in the _Rabdologia_, and in which the
logarithm of unity is 0. He also mentions that he has published some
remarks upon the propositions in spherical trigonometry and upon the new
species of logarithms by Henry Briggs, "qui novi hujus Canonis
supputandi laborem gravissimum, pro singulari amicitiâ quae illi cum
Patre meo L. M. intercessit, animo libentissimo in se suscepit; creandi
methodo, et usuum explanatione Inventori relictis. Nunc autem ipso ex
hâc vitâ evocato, totius negotii onus doctissimi Briggii humeris
incumbere, et Sparta haec ornanda illi sorte quadam obtigisse videtur."
In the address prefixed to the _Arithmetica logarithmica_ (1625) Briggs
bids the reader not to be surprised that these logarithms are different
from those published in the _Descriptio_:--
"Ego enim, cum meis auditoribus Londini, publice in Collegio
Greshamensi horum doctrinam explicarem; animadverti multo futurum
commodius, si Logarithmus sinus totius servaretur 0 (ut in Canone
mirifico), Logarithmus autem partis decimae ejusdem sinus totius,
nempe sinus 5 graduum, 44, m. 21, s., esset 10000000000. atque ea de
re scripsi statim ad ipsum authorem, et quamprimum per anni tempus, et
vacationem a publico docendi munere licuit, profectus sum Edinburgum;
ubi humanissime ab eo acceptus haesi per integrum mensem. Cum autem
inter nos de horum mutatione sermo haberetur; ille se idem dudum
sensisse, et cupivisse dicebat: veruntamen istos, quos jam paraverat
edendos curasse, donec alios, si per negotia et valetudinem liceret,
magis commodos confecisset. Istam autem mutationem ita faciendam
censebat, ut 0 esset Logarithmus unitatis, et 10000000000 sinus
totius: quod ego longe commodissimum esse non potui non agnoscere.
Coepi igitur, ejus hortatu, rejectis illis quos anteà paraveram, de
horum calculo serio cogitare; et sequenti aestate iterum profectus
Edinburgum, horum quos hic exhibeo praecipuos, illi ostendi, idem
etiam tertia aestate libentissime facturus, si Deus illum nobis tamdiu
superstitem esse voluisset."
There is also a reference to the change of the logarithms on the
title-page of the work.
These extracts contain all the original statements made by Napier,
Robert Napier and Briggs which have reference to the origin of decimal
logarithms. It will be seen that they are all in perfect agreement.
Briggs pointed out in his lectures at Gresham College that it would be
more convenient that 0 should stand for the logarithm of the whole sine
as in the _Descriptio_, but that the logarithm of the tenth part of the
whole sine should be 10,000,000,000. He wrote also to Napier at once;
and as soon as he could he went to Edinburgh to visit him, where, as he
was most hospitably received by him, he remained for a whole month. When
they conversed about the change of system, Napier said that he had
perceived and desired the same thing, but that he had published the
tables which he had already prepared, so that they might be used until
he could construct others more convenient. But he considered that the
change ought to be so made that 0 should be the logarithm of unity and
10,000,000,000 that of the whole sine, which Briggs could not but admit
was by far the most convenient of all. Rejecting therefore, those which
he had prepared already, Briggs began, at Napier's advice, to consider
seriously the question of the calculation of new tables. In the
following summer he went to Edinburgh and showed Napier the principal
portion of the logarithms which he published in 1624. These probably
included the logarithms of the first chiliad which he published in 1617.
It has been thought necessary to give in detail the facts relating to
the conversion of the logarithms, as unfortunately Charles Hutton in his
history of logarithms, which was prefixed to the early editions of his
_Mathematical Tables_, and was also published as one of his
_Mathematical Tracts_, has charged Napier with want of candour in not
telling the world of Briggs's share in the change of system, and he
expresses the suspicion that "Napier was desirous that the world should
ascribe to him alone the merit of this very useful improvement of the
logarithms." According to Hutton's view, the words, "_it is to be hoped_
that his posthumous work" ... which occur in the preface to the
_Chilias_, were a modest hint that the share Briggs had had in changing
the logarithms should be mentioned, and that, as no attention was paid
to it, he himself gave the account which appears in the _Arithmetica_ of
1624. There seems, however, no ground whatever for supposing that Briggs
meant to express anything beyond his hope that the reason for the
alteration would be explained in the posthumous work; and in his own
account, written seven years after Napier's death and five years after
the appearance of the work itself, he shows no injured feeling whatever,
but even goes out of his way to explain that he abandoned his own
proposed alteration in favour of Napier's, and, rejecting the tables he
had already constructed, began to consider the calculation of new ones.
The facts, as stated by Napier and Briggs, are in complete accordance,
and the friendship existing between them was perfect and unbroken to the
last. Briggs assisted Robert Napier in the editing of the "posthumous
work," the _Constructio_, and in the account he gives of the alteration
of the logarithms in the _Arithmetica_ of 1624 he seems to have been
more anxious that justice should be done to Napier than to himself;
while on the other hand Napier received Briggs most hospitably and
refers to him as "amico mihi longè charissimo."
Hutton's suggestions are all the more to be regretted as they occur as a
history which is the result of a good deal of investigation and which
for years was referred to as an authority by many writers. His prejudice
against Napier naturally produced retaliation, and Mark Napier in
defending his ancestor has fallen into the opposite extreme of
attempting to reduce Briggs to the level of a mere computer. In
connexion with this controversy it should be noticed that the
"Admonitio" on the last page of the _Descriptio_, containing the
reference to the new logarithms, does not occur in all the copies. It is
printed on the back of the last page of the table itself, and so cannot
have been torn out from the copies that are without it. As there could
have been no reason for omitting it after it had once appeared, we may
assume that the copies which do not have it are those which were first
issued. It is probable, therefore, that Briggs's copy contained no
reference to the change, and it is even possible that the "Admonitio"
may have been added after Briggs had communicated with Napier. As
special attention has not been drawn to the fact that some copies have
the "Admonitio" and some have not, different writers have assumed that
Briggs did or did not know of the promise contained in the "Admonitio"
according as it was present or absent in the copies they had themselves
referred to, and this has given rise to some confusion. It may also be
remarked that the date frequently assigned to Briggs's first visit to
Napier is 1616, and not 1615 as stated above, the reason being that
Napier was generally supposed to have died in 1618 until Mark Napier
showed that the true date was 1617. When the _Descriptio_ was published
Briggs was fifty-seven years of age, and the remaining seventeen years
of his life were devoted with steady enthusiasm to extend the utility of
Napier's great invention.
The only other mathematician besides Napier who grasped the idea on
which the use of logarithm depends and applied it to the construction of
a table is Justus Byrgius (Jobst Bürgi), whose work _Arithmetische und
geometrische Progress-Tabulen_ ... was published at Prague in 1620, six
years after the publication of the _Descriptio_ of Napier. This table
distinctly involves the principle of logarithms and may be described as
a modified table of antilogarithms. It consists of two series of
numbers, the one being an arithmetical and the other a geometrical
progression: thus
0, 1,0000 0000
10, 1,0001 0000
20, l,0002 0001
. . . .
990, l,0099 4967
. . . .
In the arithmetical column the numbers increase by 10, in the
geometrical column each number is derived from its predecessor by
multiplication by 1.0001. Thus the number 10x in the arithmetical column
corresponds to 10^8 (1.0001)^x in the geometrical column; the
intermediate numbers being obtained by interpolation. If we divide the
numbers in the geometrical column by 10^8 the correspondence is between
10x and (1.0001)^x, and the table then becomes one of antilogarithms,
the base being (1.0001)^{1/10}, viz. for example (l.0001)^{1/10·990} =
1.00994967. The table extends to 230270 in the arithmetical column, and
it is shown that 230270.022 corresponds to 9.9999 9999 or 109 in the
geometrical column; this last result showing that (1.0001)^{23027.022} =
10. The first contemporary mention of Byrgius's table occurs on page 11
of the "Praecepta" prefixed to Kepler's _Tabulae Radolphinae_ (1627);
his words are: "apices logistici J. Byrgio multis annis ante editionem
Neperianam viam praeiverent ad hos ipsissimos logarithmos. Etsi homo
cunctator et secretorum suorum custos foetum in partu destituit, non ad
usus publicos educavit." Another reference to Byrgius occurs in a work
by Benjamin Bramer, the brother-in-law and pupil of Byrgius, who,
writing in 1630, says that the latter constructed his table twenty years
ago or more.[4]
As regards priority of publication, Napier has the advantage by six
years, and even fully accepting Bramer's statement, there are grounds
for believing that Napier's work dates from a still earlier period.
The power of 10, which occurs as a factor in the tables of both Napier
and Byrgius, was rendered necessary by the fact that the decimal point
was not yet in use. Omitting this factor in the case of both tables,
the connexion between N a number and L its "logarithm" is
N = (e^-1)^L (Napier), L =(1.0001)^[(1/10)N] (Byrgius),
viz. Napier gives logarithms to base e^{-1}, Byrgius gives
antilogarithms to base (1.0001)^{1/10}.
There is indirect evidence that Napier was occupied with logarithms as
early as 1594, for in a letter to P. Crügerus from Kepler, dated
September 9, 1624 (Frisch's _Kepler_, vi. 47), there occurs the
sentence: "Nihil autem supra Neperianam rationem esse puto: etsi quidem
Scotus quidam literis ad Tychonem 1594 scriptis jam spem fecit Canonis
illius Mirifici." It is here distinctly stated that some Scotsman in the
year 1594, in a letter to Tycho Brahe, gave him some hope of the
logarithms; and as Kepler joined Tycho after his expulsion from the
island of Huen, and had been so closely associated with him in his work,
he would be likely to be correct in any assertion of this kind. In
connexion with Kepler's statement the following story, told by Anthony
wood in the _Athenae Oxonienses_, is of some importance:--
"It must be now known, that one Dr Craig, a Scotchman ... coming out
of Denmark into his own country, called upon Joh. Neper, Baron of
Mercheston, near Edinburgh, and told him, among other discourses, of a
new invention in Denmark (by Longomontanus, as 'tis said), to save the
tedious multiplication and division in astronomical calculations.
Neper being solicitous to know farther of him concerning this matter,
he could give no 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 showed him a rude draught of what he called _Canon mirabilis
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."
This story, though obviously untrue in some respects, gives valuable
information by connecting Dr Craig with Napier and Longomontanus, who
was Tycho Brahe's assistant. Dr Craig was John Craig, the third son of
Thomas Craig, who was one of the colleagues of Sir Archibald Napier,
John Napier's father, in the office of justice-depute. Between John
Craig and John Napier a friendship sprang up which may have been due to
their common taste for mathematics. There are extant three letters from
Dr John Craig to Tycho Brahe, which show that he was on the most
friendly terms with him. In the first letter, of which the date is not
given, Craig says that Sir William Stuart has safely delivered to him,
"about the beginning of last winter," the book which he sent him. Now
Mark Napier found in the library of the university of Edinburgh a
mathematical work bearing a sentence in Latin which he translates, "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 written this at Uraniburg, 2d
November 1588." As Sir William Stuart was sent to Denmark to arrange the
preliminaries of King James's marriage, and returned to Edinburgh on the
15th of November 1588, it would seem probable that this was the volume
referred to by Craig. It appears from Craig's letter, to which we may
therefore assign the date 1589, that, five years before, he had made an
attempt to reach Uranienburg, but had been baffled by the storms and
rocks of Norway, and that ever since then he had been longing to visit
Tycho. Now John Craig was physician to the king, and in 1590 James VI.
spent some days at Uranienburg, before returning to Scotland from his
matrimonial expedition. It seems not unlikely therefore that Craig may
have accompanied the king in his visit to Uranienburg.[5] In any case it
is certain that Craig was a friend and correspondent of Tycho's, and it
is probable that he was the "Scotus quidam."
We may infer therefore that as early as 1594 Napier had communicated to
some one, probably John Craig, his hope of being able to effect a
simplification in the processes of arithmetic. Everything tends to show
that the invention of logarithms was the result of many years of labour
and thought,[6] undertaken with this special object, and it would seem
that Napier had seen some prospect of success nearly twenty years before
the publication of the _Descriptio_. It is very evident that no mere
hint with regard to the use of proportional numbers could have been of
any service to him, but it is possible that the news brought by Craig of
the difficulties placed in the progress of astronomy by the labour of
the calculations may have stimulated him to persevere in his efforts.
The "new invention in Denmark" to which Anthony Wood refers as having
given the hint to Napier was probably the method of calculation called
prosthaphaeresis (often written in Greek letters [Greek:
prosthaphairesis]), which had its origin in the solution of spherical
triangles.[7] The method consists in the use of the formula
sin a sin b = ½{cos (a - b) - cos (a + b)},
by means of which the multiplication of two sines is reduced to the
addition or subtraction of two tabular results taken from a table of
sines; and, as such products occur in the solution of spherical
triangles, the method affords the solution of spherical triangles in
certain cases by addition and subtraction only. It seems to be due to
Wittich of Breslau, who was assistant for a short time to Tycho Brahe;
and it was used by them in their calculations in 1582. Wittich in 1584
made known at Cassel the calculation of one case by this
prosthaphaeresis; and Justus Byrgius proved it in such a manner that
from his proof the extension to the solution of all triangles could be
deduced.[8] Clavius generalized the method in his treatise _De
astrolabio_ (1593), lib. i. lemma liii. The lemma is enunciated as
follows:--
"Quaestiones omnes, quae per sinus, tangentes, atque secantes absolvi
solent, per solam prosthaphaeresim, id est, per solam additionem,
subtractionem, sine laboriosa numerorum multiplicatione divisioneque
expedire."
Clavius then refers to a work of Raymarus Ursus Dithmarsus as containing
an account of a particular case. The work is probably the _Fundamentum
astronomicum_ (1588). Longomontanus, in his _Astronomia Danica_ (1622),
gives an account of the method, stating that it is not to be found in
the writings of the Arabs or Regiomontanus. As Longomontanus is
mentioned in Anthony Wood's anecdote, and as Wittich as well as
Longomontanus were assistants of Tycho, we may infer that Wittich's
prosthaphaeresis is the method referred to by Wood.
It is evident that Wittich's prosthaphaeresis could not be a good method
of practically effecting multiplications unless the quantities to be
multiplied were sines, on account of the labour of the interpolations.
It satisfies the condition, however, equally with logarithms, of
enabling multiplication to be performed by the aid of a table of single
entry; and, analytically considered, it is not so different in principle
from the logarithmic method. In fact, if we put xy = [phi](X + Y), X
being a function of x only and Y a function of y only, we can show that
we must have X = Ae^(qx), y = Be^(qy); and if we put xy = [phi](X + Y) -
[phi](X - Y), the solutions are [phi](X + Y) = ¼(x + y)², and x = sin X,
y = sin Y, [phi](X + Y) = -½cos(X + Y). The former solution gives a
method known as that of quarter-squares; the latter gives the method of
prosthaphaeresis.
An account has now been given of Napier's invention and its publication,
the transition to decimal logarithms, the calculation of the tables by
Briggs, Vlacq and Gunter, as well as of the claims of Byrgius and the
method of prosthaphaeresis. To complete the early history of logarithms
it is necessary to return to Napier's _Descriptio_ in order to describe
its reception on the continent, and to mention the other logarithmic
tables which were published while Briggs was occupied with his
calculations.
John Kepler, who has been already quoted in connexion with Craig's visit
to Tycho Brahe, received the invention of logarithms almost as
enthusiastically as Briggs. His first mention of the subject occurs in a
letter to Schikhart dated the 11th of March 1618, in which he
writes-"Extitit Scotus Baro, cujus nomen mihi excidit, qui praeclari
quid praestitit, necessitate omni multiplicationum et divisionum in
meras additiones et subtractiones commutata, nec sinibus utitur; at
tamen opus est ipsi tangentium canone: et varietas, crebritas,
difficultasque additionum subtractionumque alicubi laborem multiplicandi
et dividendi superat." This erroneous estimate was formed when he had
seen the _Descriptio_ but had not read it; and his opinion was very
different when he became acquainted with the nature of logarithms. The
dedication of his _Ephemeris_ for 1620 consists of a letter to Napier
dated the 28th of July 1619, and he there congratulates him warmly on
his invention and on the benefit he has conferred upon astronomy
generally and upon Kepler's own Rudolphine tables. He says that,
although Napier's book had been published five years, he first saw it at
Prague two years before; he was then unable to read it, but last year he
had met with a little work by Benjamin Ursinus[9] containing the
substance of the method, and he at once recognized the importance of
what had been effected. He then explains how he verified the canon, and
so found that there were no essential errors in it, although there were
a few inaccuracies near the beginning of the quadrant, and he proceeds,
"Haec te obiter scire volui, ut quibus tu methodis incesseris, quas non
dubito et plurimas et ingeniosissimas tibi in promptu esse, eas publici
juris fieri, mihi saltem (puto et caeteris) scires fore gratissimum;
eoque percepto, tua promissa folio 57, in debitum cecidisse
intelligeres." This letter was written two years after Napier's death
(of which Kepler was unaware), and in the same year as that in which the
_Constructio_ was published. In the same year (1620) Napier's
_Descriptio_ (1614) and _Constructio_ (1619) were reprinted by
Bartholomew Vincent at Lyons and issued together.[10]
Napier calculated no logarithms of numbers, and, as already stated, the
logarithms invented by him were not to base e. The first logarithms to
the base e were published by John Speidell in his _New Logarithmes_
(London, 1619), which contains hyperbolic log sines, tangents and
secants for every minute of the quadrant to 5 places of decimals.
In 1624 Benjamin Ursinus published at Cologne a canon of logarithms
exactly similar to Napier's in the _Descriptio_ of 1614, only much
enlarged. The interval of the arguments is 10´´, and the results are
given to 8 places; in Napier's canon the interval is 1', and the number
of places is 7. The logarithms are strictly Napierian, and the
arrangement is identical with that in the canon of 1614. This is the
largest Napierian canon that has ever been published.
In the same year (1624) Kepler published at Marburg a table of Napierian
logarithms of sines with certain additional columns to facilitate
special calculations.
The first publication of Briggian logarithms on the continent is due to
Wingate, who published at Paris in 1625 his _Arithmétique
logarithmétique_, containing seven-figure logarithms of numbers up to
1000, and log sines and tangents from Gunter's _Canon_ (1620). In the
following year, 1626, Denis Henrion published at Paris a _Traicté des
Logarithmes_, containing Briggs's logarithms of numbers up to 20,001 to
10 places, and Gunter's log sines and tangents to 7 places for every
minute. In the same year de Decker also published at Gouda a work
entitled _Nieuwe Telkonst, inhoudende de Logarithmi voor de Ghetallen
beginnende van 1 tot 10,000_, which contained logarithms of numbers up
to 10,000 to 10 places, taken from Briggs's _Arithmetica_ of 1624, and
Gunter's log sines and tangents to 7 places for every minute.[11] Vlacq
rendered assistance in the publication of this work, and the privilege
is made out to him.
The invention of logarithms and the calculation of the earlier tables
form a very striking episode in the history of exact science, and, with
the exception of the _Principia_ of Newton, there is no mathematical
work published in the country which has produced such important
consequences, or to which so much interest attaches as to Napier's
_Descriptio_. The calculation of tables of the natural trigonometrical
functions may be said to have formed the work of the last half of the
16th century, and the great canon of natural sines for every 10 seconds
to 15 places which had been calculated by Rheticus was published by
Pitiscus only in 1613, the year before that in which the _Descriptio_
appeared. In the construction of the natural trigonometrical tables
Great Britain had taken no part, and it is remarkable that the discovery
of the principles and the formation of the tables that were to
revolutionize or supersede all the methods of calculation then in use
should have been so rapidly effected and developed in a country in which
so little attention had been previously devoted to such questions.
For more detailed information relating to Napier, Briggs and Vlacq,
and the invention of logarithms, the reader is referred to the life of
Briggs in Ward's _Lives of the Professors of Gresham College_ (London,
1740); Thomas Smith's _Vitae quorundam eruditissimorum et illustrium
virorum_ (Vita Henrici Briggii) (London, 1707); Mark Napier's _Memoirs
of John Napier_ already referred to, and the same author's _Naperi
libri qui supersunt_ (1839); Hutton's _History_; de Morgan's article
already referred to; Delambre's _Histoire de l'Astronomie moderne_;
the report on mathematical tables in the _Report of the British
Association_ for 1873; and the _Philosophical Magazine_ for October
and December 1872 and May 1873. It may be remarked that the date
usually assigned to Briggs's first visit to Napier is 1616 and not
1615 as stated above, the reason being that Napier was generally
supposed to have died in 1618; but it was shown by Mark Napier that
the true date is 1617.
In the years 1791-1807 Francis Maseres published at London, in six
volumes quarto "Scriptores Logarithmici, or a collection of several
curious tracts on the nature and construction of logarithms, mentioned
in Dr Hutton's historical introduction to his new edition of Sherwin's
mathematical tables ...," which contains reprints of Napier's
_Descriptio_ of 1614, Kepler's writings on logarithms (1624-1625), &c.
In 1889 a translation of Napier's _Constructio_ of 1619 was published by
Walter Rae Macdonald. Some valuable notes are added by the translator,
in one of which he shows the accuracy of the method employed by Napier
in his calculations, and explains the origin of a small error which
occurs in Napier's table. Appended to the Catalogue is a full and
careful bibliography of all Napier's writings, with mention of the
public libraries, British and foreign, which possess copies of each. A
facsimile reproduction of Bartholomew Vincent's Lyons edition (1620) of
the _Constructio_ was issued in 1895 by A. Hermann at Paris (this
imprint occurs on page 62 after the word "Finis").
It now remains to notice briefly a few of the more important events in
the history of logarithmic tables subsequent to the original
calculations.
_Common or Briggian Logarithms of Numbers._--Nathaniel Roe's _Tabulae
logarithmicae_ (1633) was the first complete seven-figure table that
was published. It contains seven-figure logarithms of numbers from 1
to 100,000, with characteristics unseparated from the mantissae, and
was formed from Vlacq's table (1628) by leaving out the last three
figures. All the figures of the number are given at the head of the
columns, except the last two, which run down the extreme columns--1 to
50 on the left-hand side, and 50 to 100 on the right-hand side. The
first four figures of the logarithms are printed at the top of the
columns. There is thus an advance half way towards the arrangement now
universal in seven-figure tables. The final step was made by John
Newton in his _Trigonometria Britannica_ (1658), a work which is also
noticeable as being the only extensive eight-figure table that until
recently had been published; it contains logarithms of sines, &c., as
well as logarithms of numbers.
In 1705 appeared the original edition of Sherwin's tables, the first
of the series of ordinary seven-figure tables of logarithms of numbers
and trigonometrical functions such as are in general use now. The work
went through several editions during the 18th century, and was at
length superseded in 1785 by Hutton's tables, which continued in
successive editions to maintain their position for a century.
In 1717 Abraham Sharp published in his _Geometry Improv'd_ the
Briggian logarithms of numbers from 1 to 100, and of primes from 100
to 1100, to 61 places; these were copied into the later editions of
Sherwin and other works.
In 1742 a seven-figure table was published in quarto form by Gardiner,
which is celebrated on account of its accuracy and of the elegance of
the printing. A French edition, which closely resembles the original,
was published at Avignon in 1770.
In 1783 appeared at Paris the first edition of François Callet's
tables, which correspond to those of Hutton in England. These tables,
which form perhaps the most complete and practically useful collection
of logarithms for the general computer that has been published, passed
through many editions.
In 1794 Vega published his _Thesaurus logarithmorum completus_, a
folio volume containing a reprint of the logarithms of numbers from
Vlacq's _Arithmetica logarithmica_ of 1628, and _Trigonometria
artificialis_ of 1633. The logarithms of numbers are arranged as in an
ordinary seven-figure table. In addition to the logarithms reprinted
from the _Trigonometria_, there are given logarithms for every second
of the first two degrees, which were the result of an original
calculation. Vega devoted great attention to the detection and
correction of the errors in Vlacq's work of 1628. Vega's _Thesaurus_
has been reproduced photographically by the Italian government. Vega
also published in 1797, in 2 vols. 8vo, a collection of logarithmic
and trigonometrical tables which has passed through many editions, a
very useful one volume stereotype edition having been published in
1840 by Hülsse. The tables in this work may be regarded as to some
extent supplementary to those in Callet.
If we consider only the logarithms of numbers, the main line of
descent from the original calculation of Briggs and Vlacq is Roe, John
Newton, Sherwin, Gardiner; there are then two branches, viz. Hutton
founded on Sherwin and Callet on Gardiner, and the editions of Vega
form a separate offshoot from the original tables. Among the most
useful and accessible of modern ordinary seven-figure tables of
logarithms of numbers and trigonometrical functions may be mentioned
those of Bremiker, Schrön and Bruhns. For logarithms of numbers only
perhaps Babbage's table is the most convenient.[12]
In 1871 Edward Sang published a seven-figure table of logarithms of
numbers from 20,000 to 200,000, the logarithms between 100,000 and
200,000 being the result of a new calculation. By beginning the table
at 20,000 instead of at 10,000 the differences are halved in
magnitude, while the number of them in a page is quartered. In this
table multiples of the differences, instead of proportional parts, are
given.[13] John Thomson of Greenock (1782-1855) made an independent
calculation of logarithms of numbers up to 120,000 to 12 places of
decimals, and his table has been used to verify the errata already
found in Vlacq and Briggs by Lefort (see _Monthly Not. R.A.S._ vol.
34, p. 447). A table of ten-figure logarithms of numbers up to 100,009
was calculated by W. W. Duffield and published in the _Report of the
U.S. Coast and Geodetic Survey for 1895-1896_ as Appendix 12, pp.
395-722. The results were compared with Vega's _Thesaurus_ (1794)
before publication.
_Common or Briggian Logarithms of Trigonometrical Functions._--The
next great advance on the Trigonometria artificialis took place more
than a century and a half afterwards, when Michael Taylor published in
1792 his seven-decimal table of log sines and tangents to every second
of the quadrant; it was calculated by interpolation from the
_Trigonometria_ to 10 places and then contracted to 7. On account of
the great size of this table, and for other reasons, it never came
into very general use, Bagay's _Nouvelles tables astronomiques_
(1829), which also contains log sines and tangents to every second,
being preferred; this latter work, which for many years was difficult
to procure, has been reprinted with the original title-page and date
unchanged. The only other logarithmic canon to every second that has
been published forms the second volume of Shortrede's _Logarithmic
Tables_ (1849). In 1784 the French government decided that new tables
of sines, tangents, &c., and their logarithms, should be calculated in
relation to the centesimal division of the quadrant. Prony was charged
with the direction of the work, and was expressly required "non
seulement à composer des tables qui ne laissassent rien à désirer
quant à l'exactitude, mais à en faire le monument de calcul le plus
vaste et le plus imposant qui eût jamais été exécuté ou même conçu."
Those engaged upon the work were divided into three sections: the
first consisted of five or six mathematicians, including Legendre, who
were engaged in the purely analytical work, or the calculation of the
fundamental numbers; the second section consisted of seven or eight
calculators possessing some mathematical knowledge; and the third
comprised seventy or eighty ordinary computers. The work, which was
performed wholly in duplicate, and independently by two divisions of
computers, occupied two years. As a consequence of the double
calculation, there are two manuscripts, one deposited at the
Observatory, and the other in the library of the Institute, at Paris.
Each of the two manuscripts consists essentially of seventeen large
folio volumes, the contents being as follows:--
Logarithms of numbers up to 200,000 8 vols.