Preliminary Manual of:
TA (TABLE-ANALYSIS version 1.1) and
MM (Very Useful Program for Analysing MEAN MOTION Tables)
by Benno van Dalen
The programs TA (Table Analysis) and MM (Very Useful Program for
Analysing Mean-Motion Tables) are designed for entering, editing,
printing and analysing astronomical tables in the Ptolemaic
tradition. The two programs operate essentially in the same way
and have basically the same set of commands. However, MM is
limited to mean motion tables, whereas TA can deal with most of
the remaining types of Ptolemaic astronomical tables including
those for trigonometric functions, spherical astronomy and the
planetary models. Both programs offer the possibility to enter
tables from manuscripts in a convenient way and to store these
tables on disk. The tables can be analysed mathematically and
their underlying parameters can be estimated in various ways. In
addition, the tables can be recomputed, compared, plotted and
printed on the screen or on a printer. TA has some fancy commands
that allow the user to compute practically any type of table he
may want to compute.
A description of the commands in both programs is found below
in the sections "GENERAL INFORMATION", "COMMANDS in MM" and
"COMMANDS in TA". Most of the commands are self-explanatory; they
will explicitly request all information needed to perform them
correctly. Some fancier or "invisible" features are explained
in this preliminary manual or on the help screens of TA (see
below). General information about the input of sexagesimal
numbers, filenames, strings, etc. is given in the section
"GENERAL INPUT".
TA has a context-sensitive help system, which can be invoked at
any time by pressing the function key F1. At the TABLE-ANALYSIS>
prompt F1 supplies general information about TA and about groups
of commands. While performing a particular command, the help system
displays various screens with information about that command
and about the particular input that is expected. You can switch
between these screens by typing the UP and DOWN ARROW keys. By
pressing the ESC key you can leave the help system and return to
the command you were performing. Pressing F1 twice gives you help
on the help system.
GENERAL INFORMATION
In order to run TA, you need the following files: TA.EXE, TA.HLP,
TYPEMENU.DAT and a TurboPascal graphics driver, which has the
extension .BGI (for example EGAVGA.BGI for a VGA screen). TA.HLP
contains all information that is displayed by the on-line help
system invoked by the function key F1. TYPEMENU.DAT contains the
complete list of types of tables that TA can handle; this list is
displayed page by page whenever you type a question mark at the
"Which type of table?" prompt. The graphics driver for your
particular screen is necessary for displaying plots of tables
or confidence regions for a least squares estimation.
All four files should be in the same directory. The configuration
file TA.CFG, which need not be present at start-up, will always
be read from and written to the current directory. This makes
it possible to maintain different configurations in different
directories. In the configuration file the colours of the screen
specified by means of the command CC (Change Colours) and "history"
values for file names, names of external zijes, directories and
formulae are saved.
In order ro run MM, you only need the executable file MM.EXE.
[Remark: the beta version 1.1 of MM includes some graphical
displays which, like TA, require a .BGI driver file.]
INTERNAL AND EXTERNAL ZIJES
Every table entered or computed in TA or MM is a "record" (block of
information) which completely describes the table: the range of the
arguments, all tabular values, name, type, underlying parameter
values, indication whether the table has been saved on disk, etc.
Every table that is entered or created in TA or MM is added to
the so-called INTERNAL ZIJ. This is a set of tables which is
always readily available to the user and on which a large number
of commands can be performed. You can obtain a numbered list of
the tables present in the internal zij by typing the command LT
(List Tables).
[Remark: TA has a second command LD (List tables plus Descriptions),
which also displays the description included in every table.]
CP (Copy) makes identical copies of a range of tables in the
internal zij, DT (Delete Tables) deletes a range of tables, MV
(MoVe; only available in the program TA) moves a range of tables,
and RN (ReNumber) renumbers the internal zij so that no gaps in
the numbering remain after some tables have been deleted or
moved. DA (Delete All tables) deletes all tables in the internal
zij; it will issue a warning if some of the tables have not been
saved to disk (see the next paragraph).
If you leave TA or MM (using the command Q for Quit), the internal
zij will be lost. Therefore you will usually want to save the tables
in the internal zij on hard drive or floppy disk using one of the
commands ST (Save Tables) or SZ (Save Zij). Both commands request
a filename; you may type any legal filename without extension, like
A:\ZIJES\BAGHDADI or \C:SHAMIL. ST also requests the range of tables
in the internal zij that must be saved; SZ simply writes all tables
in the internal zij to disk. The resulting file on disk will be
called an EXTERNAL ZIJ. The external zijes of TA all have the
extension .ZIJ, those of MM have the extension .MMT. You may use the
command LZ (List Zijes) to see which external zijes are present in
an arbitrary directory, and DZ (Delete Zij) to delete an external
zij. Using the command GT (Get Tables) you can load a range of
tables from an external zij into the internal zij, using GZ you can
load a complete external zij at once. In TA, you can use MD to Make
a new Directory, CD to Change the current Directory, and RZ to
Rename a Zij.
COMMANDS in MM
ENTERING A TABLE FROM A MANUSCRIPT.
Use the command IT (Input Table) to enter a table from a manuscript.
MM first requests some general information: the name of the table,
the type of calendar, and the type of mean motion table. Enter the
type of mean motion table carefully; the program needs the indicated
approximate mean motion per day to perform all necessary calculations
correctly.
Next the subtable menu is displayed. The types of subtables of a
mean motion table are defined as follows (in each case n denotes the
number of values in the subtable):
Fractions: arguments are 1/2, 1/3, 1/4, ..., 1/n hours.
Hours: arguments are k, 2k, 3k, ..., nk hours
(typical examples of k are 1 and 2; nk will often
be equal to 24, for some tables to 60).
Days: arguments are 1, 2, 3, ..., n days.
Months: arguments are Farvardin, Urdibihist, ... for the
Persian calendar; Muharram, Safar, ... for the
Arabic calendar; etc.
Extended years: arguments are 1, 2, 3, ..., n years.
Single years: arguments are k, 2k, 3k, ..., nk years
(typical examples of k are 10, 20, 30 and 100;
for every mean motion table in MM two subtables
for single years are available).
Collected years: arguments are E, E+k, E+2k, E+3k, ..., E+nk, where
E is the so-called epoch of the table (the
tabular value for argument E will be referred
to as "epoch value"; k will often be equal to
the number of values in the subtable for extended
years, but could for instance also be equal to 1).
After you have chosen one of these types of subtables, MM requires
the format of the subtable: the set of arguments and the number of
sexagesimal fractional digits. Next it requests the most significant
(i.e. usually the last) value in the subtable, from which it
calculates a preliminary estimate of the underlying mean motion
parameter. Using this estimate MM predicts all remaining values in
the subtable. The predicted values are displayed and can be edited
in order to make them equal to the values in the manuscript. To this
purpose you can use the + and - keys to increase or decrease the
predicted value by a single unit, the arrow keys to move the cursor,
and the digits 0 to 9 to overwrite predicted digits. Scribal errors
can be corrected and the corrections stored along with the manuscript
values: if you type C the presently displayed value will be stored
separately as the Corrected value. The corrected values will be used
for the estimation of the underlying mean motion parameter, the
manuscript values can for instance be used for an edition of the
table.
Systematic differences between the predicted values and those in
your manuscript could have various causes: an error in the tabular
value that you have supplied, an error in the specification of the
subtable concerned (e.g. an incorrect number of tabular values), or
an error in the approximate mean motion per day (or, equivalently,
in the type of mean motion table). All these possibilities can make
it impossible for MM to predict the tabular values correctly. The
PGUP key places you back on the top half of the screen and gives the
possibility to change the specification of the table or subtable
without having to start all over. Use the UP and DOWN ARROW keys to
step through the various items of the specification. Use the TAB
key or the first letter of a calendar name to change the type of
calendar; use the TAB key to change the intercalation; all other
items are edited in the way you would expect. Type PGDN to return to
the bottom half of the screen; the program gives you the possibility
to have some or all of the predicted values recomputed.
Incidental differences between predicted values and those in your
manuscript could be caused by scribal errors (see above). For certain
subtables another possibility is that the type of intercalation is
incorrect. For instance, for the Arabic calendar the values for
extended years will be different depending on the location of the
leap years within each cycle of 30 years; for the Persian calendar
the values in the subtable for months will be different depending
on the location of the epagomenal days (after the 8th or after the
12th Persian month). If you encounter an incidental difference that
might be caused by an incorrect type of intercalation, try pressing
ALT-I to change the type of intercalation. Note that the indication
of the type of intercalation on the top half of the screen changes
simultaneously.
After you have entered the last tabular value of the subtable, press
RETURN to return to the subtable menu. If you want to discard the
subtable, you can type CTRL-C at any time.
You may change a table already present in the internal zij using the
Modify command MO. The subtable menu will be displayed and you can
change subtables in the same way in which you have entered them. The
program will warn you if you try to overwrite an existing subtable.
To change the general specification of the table (like the name, the
type of calendar or the approximate mean motion per day), choose an
arbitrary subtable and move to the upper part of the screen by
pressing the PGUP key. After making the changes, you can return
to the subtable menu by pressing CTRL-C. [Remark: Version 1.1 of
MM has a separate entry in the eubtable menu for this purpose.]
TABULAR DIFFERENCES, RECOMPUTATION, COMPARISON, PRINT-OUT.
The commands DF (tabular DiFferences), RC (ReComputation), CM
(CoMpare) and PR (Print) all yield the same type of output: they
display 1, 2 or 3 mean motion tables in three adjacent columns on
the screen. In the case of DF the 1st column contains the original
table and the 2nd its tabular differences. In the case of RC the 1st
column displays the original table, the 2nd column the recomputation
(displayed to one sexagesimal digit more than the original table)
and the 3rd column possible errors expressed in units of the
original table. In the case of CM the 1st and 2nd columns contain
the tables to be compared, the 3rd column the differences between
the two. In the case of PR the columns display various mean motion
tables which can be specified by the user.
For each of the four above-mentioned commands, use the first letter of
the subtable types (F, H, D, M, E, S, C) to switch between subtables.
Press ALT-P to send the subtable currently being displayed to the
printer. If you have a laser printer, you will have to press the
line feed button on the printer to have the last page of output
printed. [Remark: This may not work properly. It is preferable to
use the command WRite in version 1.1 of MM to write all subtables
of a given table to an ASCII file and edit that file with the
word-processor of your choice.]
The command RC first requests the value of the mean motion parameter
and the type of rounding to be utilized in the recomputation. You
can enter the mean motion in the base period you prefer: hour, day,
month, year, period of extended years. MM will find the intended
motion per day (provided you have supplied the correct type of mean
motion table or the correct approximate mean motion per day!) and
will perform the recomputation accordingly.
ESTIMATION OF THE UNDERLYING MEAN MOTION PARAMETER.
MM has three methods to estimate the underlying parameter of a
mean motion table. All three offer the possibility to send the
results of the estimation to the printer. SQ (SQueeze the mean
motion parameter) squeezes the parameter from the subtables by
dividing the most significant (i.e. usually the last) value of
every subtable by the number of days involved. LNE (Least Number
of Errors criterion) gives the ranges of parameter values that
minimize the number of errors in every subtable. The number of
errors is displayed at the beginning of each line; 0 errors
corresponds to a correct recomputation. Note that the results
can be significantly different depending on the type of rounding
that you specify. LS (Least Squares estimation) can be used for
subtables that contain many errors; it determines the straight
line which fits the tabular values best.
All three estimators display estimates for every subtable present.
Press an arbitrary key after the estimates for collected, single and
extended years have been displayed to see the remaining estimates.
Bear in mind that the estimates from subtables for months, days and
hours are less significant than the others.
COMMANDS in TA
At present TA includes the majority of the non-linear types of
tables occurring in Ptolemy's Almagest and many mediaeval Islamic
astronomical handbooks. A complete list of the types supported
with short descriptions and the abbreviations (maximum 4 letters
and digits) used to identify them in the program can be found in
the file TYPEMENU.DAT, which can be inspected at the DOS level or
can be called from within TA by pressing a question mark at the
"Which type of table?" prompt. The following groups of functions
are included:
1) Trigonometric functions (sine, cosine, tangent, cotangent,
versed sine, chord, secans and cosecans).
2) Functions for spherical trigonometry and timekeeping (solar
declination, right ascension, equation of daylight, oblique
ascension, hour length, length of the longest day).
3) Functions related to the solar model (solar equation for
different independent variables, method of declination, solar
velocity, true and mean solar position, equation of time for
different independent variables).
4) Functions related to the lunar model and the planetary models
(equation of center, equation of anomaly, various interpolation
functions; latitude; lunar distance; planetary stations).
5) Functions related to parallax and solar and lunar eclipses.
Whenever you want to calculate a table of one of the types supported,
you will have to specify the values of the underlying parameters. In
a number of cases these parameters may be "composed", i.e. they are
auxiliary parameters that were computed from two actual parameters
of the function concerned. For instance, for some of the planetary
equations the underlying eccentricity and radius of the epicycle
can be replaced by a single parameter, which is called a "planetary
quotient". Instead of calculating the value of a composed parameter
yourself, you may press ESC and enter the values of the two
underlying parameters separately. Composed parameters are indicated
by the sign > after their name. In version 1.1 of TA (July 1994) the
interpretation of confidence intervals for composed parameters has
not yet been implemented.
For so-called "displaced" planetary equation tables [see, for
instance H. Salam & E.S. Kennedy, "Solar and Lunar Tables in
Early Islamic Astronomy", Journal of the American Oriental Society,
pp. 496-497.] the parameters "shift" and "displacement" have been
introduced. The shift indicates the shift in argument with respect
to the "ordinary" equation; the displacement is the constant added
to the equation. For example, the solar equation table displaying
values
1 2; 6, 2 1 0; 2, 1
2 2; 8, 2 2 0; 4, 1
3 2;10, 2 instead of 3 0; 6, 2
4 2;12, 2 4 0; 8, 2
5 2;14, 2 5 0;10, 2
has shift -2 (the values were shifted backwards!) and displacement
+2;0,0. Displaced tables are stored by TA without their shift and
displacement. In version 1.1 (July 1994) some of the operations on
displaced tables have not yet been implemented. [Remark: for this
reason, it may be more convenient to enter the tables without their
shift and displacement in the first place.]
Many of the commands in TA make use of a so-called option menu.
For example:
Print in columns: Yes (Y/N)
Print on screen or line printer: Screen (S/L)
First argument: 1
Increment: 1
Last argument: 90
You can use the UP and DOWN ARROW keys to step through such a
menu and you can change entries in the menu in the way you would
expect (cf. the section "GENERAL INPUT" below). ESC restores a
default option, CTRL-C aborts the command. Press RETURN after you
have set all options as desired to perform the command.
ENTERING A TABLE FROM A MANUSCRIPT.
The command IT in the program TA is very much like the command
IT in MM. Again the user first has to specify the structure of
the table to be entered. However, he now chooses between five
possible ways of entering a table. Firstly, the tabular values
can be predicted on the basis of an estimate of the underlying
parameter values calculated from one or more particular tabular
values (option C; not possible for all types of tables), or on
the basis of the assumption that the tabular values have constant
first or second order tabular differences (option D). Secondly,
the tabular values can be read from an ASCII file having one of
the following formats:
A B C D
0;31,25 0.5 0.5236111 0.5236111 0.5 0 31 25
1; 2,50 1.0 1.0472222 1.0472222 1.0 1 2 50
1;34,15 1.5 1.5708333 1.5708333 1.5 1 34 15
2; 5,39 2.0 2.0941667 2.0941667 2.0 2 5 39
2;37, 4 2.5 2.6177778 2.6177778 2.5 2 37 4
etc. etc. etc. etc.
(option F), or they can be taken from another table in the internal
zij (option T). Finally, the tabular values can be entered by hand
(option H).
The predicted tabular values can be edited in the same way as in MM:
by pressing the + or - key you can increase or decrease a predicted
value by a single unit; using the cursor keys and the numerical
keys, you can overwrite digits of a predicted value. As in MM, two
versions of each tabular value are stored, a manuscript version and
a corrected version. While editing the predicted tabular values you
can change between the two versions by pressing C or M. You can set
a (manuscript or corrected) tabular value to UNDEFINED by pressing
the U or * key. This option can be used to indicate an illegible
value in the manuscript or to exclude an obviously incorrect value
from the estimation of the underlying parameters, which always makes
use of the corrected values.
Note that you can restore the original predicted value by pressing
ESC as long as you have not edited another tabular value. Use the
UP and DOWN ARROW, PGUP and PGDN, and CTRL-PGUP and CTRL-PGDN keys
to move through the table. Certain types of scribal errors can be
corrected by pressing combinations of the ALT key and numerical keys
(details can be found on the help screen for editing sexagesimals;
press F1 followed by the DOWN ARROW key).
Like in MM, items of tables in the internal zij can be changed by
means of the command MO. Changing the tabular values by means of
this command works precisely as described above for entering them.
The command WR is in certain respects the opposite of IT: it writes
values from a table in the internal zij to a file on disk in one of
the above-mentioned formats A to D.
PRINTING, COMPARISON, RECOMPUTATION, RESIDUALS, PLOTTING.
The command PR (PRint) displays 1 to 6 tables on the screen or sends
them to the printer. Like most commands in TA, PR is straightforward
and requests all necessary information in a clear fashion. If you
have a header displayed on the screen, press an arbitrary key to
continue. If tabular values are displayed on the screen, use the UP
and DOWN ARROW, PGUP and PGDN, and CTRL-PGUP and CTRL-PGDN keys to
scroll the table. Press ESC to return to the TABLE-ANALYSIS> prompt.
In the same way you can inspect the output of the three commands
CM (CoMpare), RC (ReCompute) and RS (Residuals). CM compares any
two tables and displays differences and difference statistics;
RC recomputes a table for particular values of the underlying
parameters; RS computes residuals for given values of the
underlying parameters, which can then be tested statistically
using the command TR (TestResiduals).
The command PL draws a plot of a table in the internal zij on
the screen. PL won't work if the correct graphics driver file
(with extension .BGI) is not present in the directory containg
the executable file TA.EXE. When the first plot has appeared on
the screen, you can type F1 to specify a second table to be drawn
in the same plot. You can type F2 to write the plot to a file in
HP LaserJet format. This file can be printed on a HP LaserJet
using the DOS command "COPY /b PRN:".
ESTIMATION OF UNDERLYING PARAMETER VALUES.
TA provides four commands for estimating unknown parameter values
in tables of one of the supported types. General information
about the application of these estimators and the interpretation
of their results can be found on the general help screens of TA
(at the TABLE-ANALYSIS> prompt, press F1, 7, 1 and use the UP and
DOWN ARROW keys to view all available information).
PE (straightforward Parameter Estimation) computes an approximation
to a single unknown parameter from a single tabular value, which it
chooses in such a way that the accuracy in the estimate is as large
as possible. The interval displayed by PE contains all parameter
values for which the chosen tabular value is correctly recomputed.
Note, however, that more accurate results can be obtained by means
of the Least Squares estimation and the Least Number of Errors
Criterion.
FE (Fourier Estimation) computes an estimate of a translation
parameter of a periodic function (e.g. the solar apogee in a
table giving the solar equation as a function of the mean or
true solar longitude). The tabulated function must be symmetric
(f(l+x) = -f(l-x) if l is the translation parameter), but need
not be known precisely. FE computes estimates of the Fourier
coefficients of the tabulated function and uses these to estimate
the translation parameter and to calculate a so-called 95 %
confidence interval, which is expected to contain the translation
parameter in 19 out of 20 cases.
[Remarks: More information about the Fourier estimator and an
extensive example of its use can be found in Benno van Dalen,
"Ancient and Mediaeval Astronomical Tables: mathematical structure
and parameter values" (doctoral thesis), Utrecht 1993, Sections
2.3 and 2.6.3; or in Benno van Dalen, "A Table for the True Solar
Longitude in the Jami` Zij", in: Ad Radices - Festband zum
50jaehrigen Bestehen des Instituts fuer Geschichte der Naturwis-
senschaften Frankfurt am Main, Stuttgart 1994, pp. 163-181.]
LS (Least Squares estimation) determines the values of the
underlying parameters in such a way that the sum of the squares
of the errors in the table is minimized. LS displays marginal 95 %
confidence intervals for all parameters estimated. If there is more
than one unknown parameter (and the correct graphics driver file
with extension .BGI is present in the directory that contains the
file TA.EXE), you can plot the joint 95 % confidence region for
every pair of unknown parameters. For certain functions, in
particular the equation of daylight, the confidence region gives
significantly more information about the unknown parameter values
than the marginal confidence intervals.
[Remark: An extensive description of the application of the least
squares estimator to a table for the equation of time and of the
interpretation of the results of the estimation can be found in:
Benno van Dalen, "Al-Khwarizmi's Astronomical Tables Revisited:
Analysis of the Equation of Time", in: From Baghdad to Barcelona
Studies in the Islamic Exact Sciences in Honour of Prof. Juan
Vernet, Barcelona 1996.]
LNE (Least Number of Errors criterion) determines the values of
a single unknown parameter for which a recomputation of the table
under consideration contains the least possible number of errors.
LNE displays the numbers of errors for various intervals of
parameter values at the beginning of each line.
Note that PE and LNE can only be used for tables having a single
unknown parameter. FE can also be applied to tables of which the
underlying function is not precisely known. Information about the
remaining "statistical commands" can be found on the help screens
of TA. These commands include SD for calculating the Standard
Deviation of a set of tabular values; CR for calculating the
CorRelation coefficient of two sets of tabular values; TR for
Testing the Residuals of a table for the properties required
to apply the above-mentioned estimators; and MC (Monte Carlo
analysis) for testing the validity of the 95 %confidence
intervals determined from the estimations.
CALCULATION OF TABLES
You can use the command CL (CaLculate) to compute tables of the
above-mentioned standard types. A table can be extended according
to the symmetry relation(s) it is supposed to satisfy by means
of the command SE (Symmetry Extend); the Symmetry can be Checked
using the command SC. The command SP (SPecial functions) allows
you to calculate tables from other tables of certain types, e.g.
the normed right ascension from a right ascension table, or the
underlying right ascension and equation of daylight from a table
for the oblique ascension. DF computes a table of finite order
tabular differences for an arbitrary table in the internal zij.
The commands IP (InterPolation), II (Inverse Interpolation) and
TC (TableCalculator) make it possible to calculate exotic types
of functions and to simulate almost every possible method of
computation that might have been used by ancient or mediaeval
astronomers.
IP allows you to perform linear or parabolic interpolation
in an arbitrary table in the internal zij. Thus you may, for
instance, compute a sine table with values for every 5 minutes of
the argument, of which only the values for integer numbers of
degrees were calculated exactly. By taking the arguments of the
interpolation from another table, you can simulate the use of
interpolation in an intermediate stage of a calculation, for
example in the determination of the tangent of previously
calculated solar declination values (which is needed for
computing the equation of daylight).
II allows you to perform linear or parabolic inverse interpolation
in any table in the internal zij and works just like IP. By means
of this command you can for instance simulate the use of inverse
interpolation in a sine table to compute arcsines.
TC allows you to specify a formula according to which you wish to
calculate a table. The formula can include arithmetic operators,
trigonometric functions and rounding or truncation at intermediate
stages of the calculation. In particular, you can use the binary
operators * / + -, the unary operators abs sin cos tan asin acos
atan sqr sqrt, the semibinary operators ro (modern rounding) tr
(truncation) sh (shift) rev (reverse), sexagesimal numbers (23;35
or 998;12,13), table indicators (T1 or T12), and so-called
specifiers.
The syntax of the semibinary operators is
operator(expression|integer),
where the expression denotes a table. In the case of the semibinary
operators ro and tr, the integer argument is the sexagesimal frac-
tional digit after which the rounding must be performed. In the
case of sh it is the constant by which all arguments of the table
determined by the expression must be increased. For rev the integer
argument is the argument of the table in which all arguments must
be mirrored. For instance, rev(T1|45) is the table T for which
T(45-x) = T1(45+x) for every x=-45,-44,...,+45.
The syntax of a specifier is
[] or [0;15|0;15|90] or [T1].
The specifier does two things. Firstly it determines the range of
arguments of the new table: [] stands for the default range 1, 2,
3, ..., 90; [0;15|0;15|90] specifies the range 0;15, 0;30, 0;45,
..., 90; [T1] duplicates the range of table 1. Secondly, when the
formula is evaluated the specifier is replaced by the value of
the argument in the first two cases and by a tabular value of
table 1 in the third case. If the specifier is placed after a
colon at the end of the formula, it only specifies the range of
arguments of the table to be calculated.
Examples:
tr(sin([0;15|0;15|90])|3)/tr(cos([])|3) (1)
5+sh(atan(2;30*sin([])/(60+2;30*cos([])))|5) (2)
T1/rev([T1]|45) (3)
T1:[T2] (4)
(1) computes a tangent table for arguments 0;15, 0;30, 0;45, ...,
90, where the intermediate sine and cosine are truncated to three
sexagesimal fractional digits. (2) yields a solar equation table
of the displayed type (see above) having arguments 1, 2, 3, ...,
90, shift 5 and displacement 5 (this table could also be computed
directly using the command CL). Assuming that table 1 in the
internal zij is a sine table, (3) yields the tangent table which
could be computed from it. (4), finally, produces a table with
the tabular values of table 1 for the range of arguments of table
2.
GENERAL INPUT
The help utility of TA, which was described above, can be invoked
by pressing F1 at any time. MM does not have an advanced help
utility, but supplies a command H or ? at the MEAN-MOTION> prompt
for displaying the list of all available commands. To return to
the prompt, type ESC or CTRL-C to abort the command you are
currently performing.
TA and MM will usually indicate what kind of input is expected, e.g.
Delete which range of tables?
Delete or Not? (D/N)
Which number of sexagesimal fractional digits or unit (0-5, U)?
Print which number of tables? (1 to 6)
Write to which file? ...............................
Illegal input will not be accepted at all, or will lead to error
messages indicating why the input was not correct.
Single key input (like D, N, U and 0 to 6 in the above examples)
will be accepted as soon as a key is pressed. As for the number of
sexagesimal fractional digits, if all tabular values of a particular
table are multiples of 0;0,4, it is advisable not to specify the
number of digits as 2, but to press U and than enter 0;0,4 as the
unit of the table. This will for instance make it easier to edit
predicted values by means of the command IT.
Multiple key input (like a range of tables, a sexagesimal number or
a filename) must be concluded by RETURN. A range of tables has the
form a-b, where a and b are numbers or void (the range - covers all
tables in the internal zij). Sexagesimal numbers can be entered in
their usual form, for instance 23;30,17 or 2s22;39 (for 82;39).
While entering a string (e.g. the name or description of a table)
you can use the CTRL-RIGHT and CTRL-LEFT ARROW keys to move the
cursor one word at a time and CTRL-T to delete a word. In TA, ALT-0
is a short-cut for the degree sign). When you enter the name of a
table, the UP ARROW key yields the previously edited name. This
could be convenient when you enter many tables with similar names
or with complicated references to a manuscript. When you enter file
names, directories or formulae (in the command TC), the UP and DOWN
ARROW keys allow you to edit the five or ten previously entered
strings.
In TA, if it is not clear what input is expected or allowed, type F1
to obtain specific help.
ADDRESS
I would much appreciate any comments or suggestions you might have
concerning either the programs TA and MM or this manual. My address
is:
Benno van Dalen
Institut fuer Geschichte der Naturwissenschaften
P.O. Box 111932 (FB 13)
60054 Frankfurt am Main
GERMANY