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Kepler and logarithmic calculation Keplers contribution
to this subject is complex, as he was plagued by quite hectic circumstances,
being between 1618 (when he became aware of Napiers invention) and his death
in 1630: -The witchcraft charge
against his mother around 1620 -The 30 year war
started in 1618 -The development of
the subject by other authors (Ursinus, Briggs, Vlacq) in quick succession in
a time with slow international communication. Kepler was under
pressure to complete the Rudolphine Tables, among others by his maecenas
Rudolph II and the Brahe heirs. As a result he was forced into decisions,
that later had to be revised in view of material newly published by other
authors. It must be said, that
Kepler always has been quite open-minded in those revisions, however to
understand both decisions and revisions a detailed analysis is clarifying. Chilias 1624
In this table the
value of sin(α) varies from 0 to 1 as α progresses from 0º to 90º.
The table, newly calculated by Kepler, has those sin values as argument with
a .001 interval. The logarithms given are positive however. As log1=0 thus in
fact the logarithms of 1/ sin(α) are given (see also Fletcher 1945,
p.179). As Kepler mentions in
a letter to Mästlin those new calculations state a value for log sin 30º of
69314.72, where Napier has 69 as the last 2 digits. The same error was
rediscovered by Sang in 1865 when comparing the Napier table with that of
Ursinus (which is also newly calculated).
The frontispiece of
the Tabula Rudolphinae shows on the roof 6 women impersonating the sciences,
of which Arithmetica has a stick in her left hand and another one in her
right hand of half lengh of the first, being the sides of a triangle relevant
for the calculation of sin 30º=0.5.The correct value of log sin 30º is shown
around her head. This curious detail does not fully apply to those tables
however, as the logarithms included are in 5 digits and the difference thus
does not show anymore. Another novelty
relates the proportions of 1 expressed in the argument column to the
corresponding value in hours, minutes and seconds/24 hours (column 3) and in
minutes and seconds/ degree (column 5). Those logistic values are of
particular use in astronomical calculations, and the further development of
this novelty in the TR must have been a major cause for the use of Keplers
tables for astronomical applications until the end of the 18th
century. Tabulae Rudolphinae 1627 The major part of
those tables is of astronomical nature, describing the motion of the sun,
moon and planets, the latitude and longitude of major cities and other basic
astronomical data. The beginning 23 pages
also give logarithmic values to ease the astronomical calculations using the
astronomical data on the later pages. It is important to be
aware, that the TR are a kind of root-tables, enabling the calculation of
annual Ephemerides, being the positions of the sun, moon and planets in terms
of actual date, time and location. The statement sometimes found, that those
logarithms have been used calculating the TR therefore is at least
incomplete, and possibly not correct. The largest part of
the logarithm section is taken by the first table titled Heptacosias
Logarithmorum Logisticorum, elaborating the concept introduced in the Chilias
of directly supplying logarithms of logistic values. Those are newly
computed, opting for intervals more appropriate for logistic values than the
.001 interval of the Chilias. In a separate table
the logarithms of sin(α) are given at a 1’ interval
(0º<α<90º) at 2 decimals less than in Napiers table. As a result
Kepler could prepare this table from that of Napier, as the error discussed
in the preceeding paragraph vanishes in the rounding. Some minor specifics
of the last decimals show, that Kepler actually used values rounded from
Napier in this table, as is illustrated by the following values of log sin:
Separate tables are
given for the logarithms of cos and tan at ranges and intervals appropriate
for use in astronomy calculations. Appendix Bartschii 1629
Bartsch completed his
appendix in 1629, and it is not present in most copies of the TR. The
following quotes come from the more complete GW. Bartsch is more practical in
advising his users on supplementary material being available in the meantime
as follows: Si vero laborem…
cupit, substituere potest Ursinus (1624), uso magno, exactissime ad dena
secunda descriptum, aut meos manuales, ex eodem cum ipsius venia derivatos,
qui paucioribus pagellis compendiose descripti,…(If it is required to work
more exactly, it can be replaced by Ursinus (1624), which is extremely useful
and at a 10 second interval, or by my Tabulae Manuales , which are extracted
from the work mentioned last with permission, and which in less pages
describe the matter in a more condensed way,..) GW X 267. Hammer states in his
Nachbericht to Keplers Gesammelte Werke (X p.51*): Without any doubt Kepler
would have been of great service to his users, in case he had decided to use
logarithms of the Briggsian type in the TR. The issue of the Kepler Heptacosias
fivefold enlarged by Jacob Bartsch in 1700 than would not have been
necessary. Bartsch and Hammer
touch on three items here, of which the significance possibly becomes more
clear, when they are dealt with seperately: Briggs 1628 Possibly Hammer is right,
however in practice this option has never been open to Kepler. Hammer himself
states (GW X 21*) that the manuscript of the TR was largely finished in
middle 1624. Briggsian logarithms of trigoniometrical functions only became
available in 1628. Ursinus 1624 Benjamin Ursinus
assisted Kepler before 1620. His Magnus Canon became available in early 1625
(the dating of the intro paragraphs), and states the logarithms of
trigoniometrical values at 3 decimals more (see above) than the TR and at a
10” internal (TR 1’). This is too late to be taken into account in the text
of the TR, completed in manuscipt in
mid 1624. The Bartsch Appendix is from 1629 however and thus can mention it
as very well supplementing the more concise logaritmic section in the TR. Also Kepler has shown
to be quite welcoming to the Magnus Canon. Ursinus owned a copy of the 1627
Kepler tables (the Honeyman #1800 copy. Owen Gingerich states that it was
acquired by the Adler Planetarium in Chicago) warmly inscribed to him, Kepler
calling him and Tycho Brahe "the scientific fathers of those
tables" (my free translation of the Latin text). As Brahe (died 1601)
contributed much to the underlying astronomical observations, for Ursinus
this remark must refer to his log tables. The provenance of our
copy of Ursinus (1624) includes the major astronomer Franz von Zach, whose
activities culminated around 1800. 175 years after publication his interest
in the book can only be understood in connection with the TR. Bartsch and Kepler
could recommend the use of Ursinus (1624) because it avoids the error in the
last 2 decimals of the Napier table, and even adds one other decimal. Bartsch, Tabulae
Manuales 1700 The significance of
this title is twofold. It summarizes Ursinus (1624) omitting the last 3 decimals, however
maintaining the 10” interval, it enlarges that of the TR. On the other hand it enlarges the Heptacosias table
fivefold, the area being of special importance for astronomical calculations,
because of the direct link of logistic values to its logarithms. Maybe this is the
major explanation of the continued use of the triad TR/Magnus Canon/ Tabulae
Manuales for 175 years, combining the astronomical data of the TR with the
most complete trigoniometrical logarithms of the Magna Canon and the smaller
interval of the Heptacosias in the Manuales. Indeed Bartsch did
complete the Tabulae Manuales at the time of his writing the 1629 Appendix to
TR, however the edition (Sagan, 1631) misfired, Bartsch dying short after its
publication, and only 2 incomplete copies (Koenigsberg and Bonn) are known
(see also Caspar 85) to survive. In 1700 Eisenschmid (a
Strassbourg mathematics professor, † 1716) corrected and republished the
Bartsch tables and added a new 9-page Praefatio. Such a reissue after 69
years can only be understood as an appreciation by contemporary astronomers
of the extension of the Heptacosias given. Evaluation
Purely from a calculus
point of view the Heptacosias in the Manuales and the logarithmic values of
trigoniometrical functions in the Magnus Canon are necessary additions to the
TR in their use for astronomical calculations, as is illustrated by their
recommendation by Kepler and Bartsch, the reissue of the Manuales as late as
1700 and the presence of the Canon in the library of Von Zach (*1754). This
makes the TR themselves less interesting within a calculus-oriented
collecting scope. Catalog
For Manuales see 33615 and for the Ursinus’ Magnus Canon 23462.
[023462]
Ursinus, Benjamin. TRIGONOMETRIA CUM MAGNO LOGARITHMOR. CANONE. Colonia: Georg Runge for Martin Gutt, 1625. First
Edition. COLLATION 4to [193*155 mm]. The 2nd part (the Canon) with a seperate
titlepage: Benjamin Ursinus, MAGNUS CANON TRIANGULORUM LOGARITHMICUS,
Colonia, Georg Runge for Martin Gutt, 1624. (4), 272, (454) pp. The 1st.
unnumbered page is the engraved title marked: Petrus Rollos fecit, the title
being on a swinging door in perspective, also described by Henderson, most
likely using the Oxford copy. Also the typo he describes (Guttij tipys) can
be found on this page. The swinging door is earlier seen on Salomon de Caus,
Les raisons des forces mouvantes, Frankfurt, Norton, 1615 (offered by
Interlibrum 253/36). The last leave has errata and a colophon dated 1624 on
the verso. The 2nd title with blank verso, thus 225 leaves remaining for the
tables (5 leaves per degree). All quires in 4, last in 3. Leave 2 and 3 are 1
sheet however, folded at spine. Most descriptions give 227 leaves for the
Canon, Macdonald (p. 158) however mentions a blank leave Lll4, which he
possibly noted in one of the UK copies. Signatures:):(4 A-Z4 Aa-LL A-Z Aa-Zz
Aaa-LLL
CONDITION Contemporary
overlapping orange (rare in that colour!) vellum (in quite good condition,
taking into account the age). In the past ages the vellum was protected from
the hands of computers by plain paper wrappers, pasted under the endpapers.
Those are now expertly removed. The wrappers are still available in a
seperate envelope, and could easily be reinstalled to a future owners taste.
Edges stained green. Some wear at corners and spine. Pages mildly browned, as
often on books from the 30 Year War period (1618-1648). Contemporary
handwritten expert notes , partly in red. The handwriting of those notes is
quite like that of Bode's name on title. PROVENANCE Johann Justus Bode, 1719
and "Albrecht", neatly inscibed in the margin of the title. V.
Zach, small stamp on dedication page. JJ Bode (1676-1719) was born in
Bodenburg. He developed a sundial for travelers. The astronomer Franz Xaver
von Zach (Pest 1754) built and supervised the Seefeld observatory for Herzog
Ernst II von Sachsen-Gotha- Altenburg in CONTENTS A copy of the most
extensive table of Napierian logarithms existing. In those log e
(2.302585...) is 1 and not log 10 as in decimal logs used universely since
1628 (the first Vlacq tables). Napier's table (1614) was to minutes in 7
decimals, extended by Ursinus to 10'' in 8 decimals, also giving 1st
differences. The tables have the following main columns: natural sines, log
sines, log tangens, log cosines and cosines, each with a D(ifference) column
behind it. As sin x=cos(90-x) the table runs to 44.59' 50", higher
values for sin being found in the corresponding cosin column and vice versa.
For easy reference those higher values are indicated at the bottom of the
page, as usual ever since. For some arcs often used even more exact values
are calculated (16 decimals, see p. 139 ff. in the text). Napier's table
essentially consists of interpolations of the "Radical Table "
developed by him (see Constructio 47). In this table the sines develop
geometrically with a factor 9995/10000 and the corresponding logarithms
develop arithmetically by the equal interval 5001.25. However in this
progression a calculating error was made, resulting in logsin 30 ending 69,
were Ursinus gives OTHER MATTERS Colonia is Kolln a d
Spree nowedays part of Berlin. The different dating of Trigonometria and
Canon suggest a Sammelband of 2 seperate issues, also the quires are lettered
seperately. KvK mentions 1 separate table (Dusseldorf) and 3 books as
described here (Cologne, Munster and Regensburg), actual copies possibly
being not all as complete as our copy. COPAC has 3 complete copies (Oxford,
BL and UCL), the collation of Oxford exactly agrees with ours. NCC has 1
complete copy (Groningen), with (autopsy 2/12/03) agrees with our collation
plus the leave Lll4 found by Macdonald in some UK copies. In conclusion, the
works are most likely issued together, an individual copy occurring
occasionally, as with the Vlacq tables. Henderson 8.0, Fletcher p. 439 and
179, Dodson III, Macdonald 157 (including a very detailed collation, in
agreement with the present copy except for a last blank leave Lll4).
Macdonald (1889) also mentions copies in Edinburgh and BNF, which cannot be
traced nowadays (2003). Not in Honeyman (never found a copy?) Quite rare and not often seen in
trade. OCLC adds 3 copies in
the US (US Naval Obs., Clark and Un. Michigan). Leave Lll4 not in standard
OCLC description, in which also collation of signatures (identical to our
copy). EVALUATION Fletcher remarks on
those tables (p. 179):" It may be noticed that the 8-decimal Napierian
canon of Ursinus 1624 has never been equalled subsequently." Further Eisenschmid
(the editor of the 1700 edition of Kepler's Tabulae Manuales, see our book
33615, and thus a much more contemporary expert than the present
bibliographer) states, that Bartsch (Keplers son in law and collaborator)
used the Ursinus table, omitting the last 3 figures. Indeed the Tabulae Manuales
are in 5 decimals. As those Tabulae Manuales are an extract of logarithms
from the Tabulae Rudolphinae (1627, which use those logs for astronomical
calculations), those famous Kepler tables are thus (according to
Eisenschmid) based on Ursinus' logarithms! In this connection it
also should be mentioned, that the cos-table in Kepler (1627) has an 10"
interval, which only can come from Ursinus, Napier having a 60"
(=minute) interval. Further Ursinus owned
a copy of the 1627 Kepler tables (the Honeyman #1800 copy. Owen Gingerich
states that it was acquired by the Adler Planetarium in Chicago) warmly
inscribed to him, Kepler calling him and Tycho Brahe "the scientific
fathers of those tables" (my free translation of the Latin text). As
Brahe (died 1601) contributed much to the underlying astronomical
observations, for Ursinus this remark must refer to his log tables. Those observations
support the statement of Eisenschmid concerning the Ursinus source of the
Kepler tables. € 7.200,00
[033615]
Kepler,
Johannes and Jakob Bartsch. TABULAE
MANUALES LOGARITHMICAE. Strassbourg:
Theodor Ierse, 1700. 40, (276) pp. a4-e4, A4-LL4, MM2. 160*95 mm., however
bound in fours. Introduction and 5 tables after seperate subtitles. 19th
century marbled boards with handwritten title on label on spine. 1cm of blank
missing at foot of title. The quite elaborate subtitle is: ad calculum
astronomicum, in specie Tabb. Rudolphinarum compendiose tractandum mire
utiles. Ob defectum prioris editionis Saganensis multum hactenus desideratae.
Quibus accessit in hac editione introductio nova curante Joh. Casp.
Eisenschmid. Kepler's Tabulae Rudolphinarum (folio, Ulm, 1627) were
astronomic tables, in which the tables of logarithms used for its calculation
were also published. The present tables are in a more practical format
(manuales), extracted from them in 1631 by Bartsch (Keplers son in law).
However the edition (Sagan, 1631) misfired, Bartsch dying short after its publication,
and only 2 incomplete copies (Koenigsberg and Bonn) are known (see also
Caspar 85). In 1700 Eisenschmid (a Strassbourg professor, died in 1716)
republished the Bartsch edited tables and added a new 9-page Praefatio. An
important copy of the first generation of logarithms (as we would say in
present day based on the natural number e) dominated by Napier in the UK and
Kepler and Ursinus on the continent. The last 2 also worked together (see
Honeyman 1800, Ursinus copy of the Tabulae of 1627). In 1628 the first
complete version of Briggsian logarithms (based on 10) was published by
Vlacq, and those were used until electronic calculators made logarithms
obsolete around 1978. Henderson 10.1 (see appendix p. 207. Henderson states
12.1, which must be incorrect as 10 are the 1627 Tabulae). 4 copies in COPAC,
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