ILLUSTRATING TIME's SHADOW the Appendices by Simon Wheaton
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ILLUSTRATING TIME’S SHADOW The Appendices by Simon Wheaton-Smith a pine woodcut print ISBN 978-0-9960026-3-9 Library of Congress Control Number: 2014904842 Simon Wheaton-Smith www.illustratingshadows.com (c) 2004-2014 Simon Wheaton-Smith All rights reserved. Jan 31, 2016 1 THE ILLUSTRATING SHADOWS COLLECTION Illustrating Shadows provides several books or booklets:- Simple Shadows Build a horizontal dial for your location. Appropriate theory. Cubic Shadows Introducing a cube dial for your location. Appropriate theory. Cutting Shadows Paper cutouts for you to make sundials with. Illustrating Times Shadow the big book Illustrating Times Shadow ~ Some 400 pages covering almost every aspect of dialing. Includes a short appendix. Appendices Illustrating Times Shadow ~ The Appendices ~ Some 180 pages of optional detailed appendix material. Supplement Supplemental Shadows ~ Material in the form of a series of articles, covers more on the kinds of time, declination confusion, other proofs for the vertical decliner, Saxon, scratch, and mass dials, Islamic prayer times (asr), dial furniture, and so on! Programming Shadows A book discussing many programming languages, their systems and how to get them, many being free, and techniques for graphical depictions. This covers the modern languages, going back into the mists of time. Legacy languages include ALGOL, FORTRAN, the IBM 1401 Autocoder and SPS, the IBM 360 assembler, and Illustrating Shadows provides simulators for them, including the source code. Then C, PASCAL, BASIC, JAVA, Python, and the Lazarus system, as well as Octave, Euler, and Scilab. And of course DeltaCAD and its Basic variant, Python as in FreeCAD and Blender CAD systems, VBS and Java Script as in NanoCAD, programming TurboCAD (VBS and parametric script), and LISP as in the ProgeCAD system. And so on! Illustrating Shadows provides a variety of software tools:- CAD DeltaCAD ~ macros for almost all dialing needs in BASIC NanoCAD ~ dial macros written in VBS and Java Script FreeCAD ~ dial macros written in Python Powerdraw ~ dial macros in a Pascal subset ProgeCAD ~ dial macros written in LISP TurboCAD ~ dial macros written in VBS, and parametric part scripts also Blender ~ dial macros written in Python Languages Programs in the languages are discussed in Programming Shadows Spreadsheets illustratingShadows.xls simpleShadows.xls cubicShadows.xls Updates Check for general updates and corrections at:- www.illustratingshadows.com/reference or scan the QR code to the left which takes you there. 2 THE APPENDICES Appendix 1 Geometry and trigonometry Trigonometric functions and geometrical rules and circular measures Tables of trig values – sin, cos, and tan Appendix 2 Tables independent of latitude or longitude Julian day Declination of the sun by the day The equation of time, generic and astronomical, and century comparisons Sun's apparent hour angle, and standard time to hour angle Longitude to time Aids to locating north, compass correction and astro compass, noon transit time Latitude, Longitude, Magnetic declination information Appendix 3 Location dependant ~ miscellaneous dial hour line data Hour line angles for horizontal and vertical dials latitude 30 to 60 Reflecting or ceiling dial information. Polar and Meridian dial tables because they are not latitude dependant Equatorial calendar line radii and sunset line distance data Appendix 4 Location dependant ~ Tables of altitude, azimuth, and calendar data Table for analemmatic dials – angle to hour point, and analemma Tables for calendar lines for flat dials lat 0-65, declinations 23.5, 0, -23.5 Conversion of style linear length and linear height for several latitudes Appendix 5 Location dependant ~ Vertical decliners, and miscellaneous useful data Mostly south facing decliner hour line angles and SD and SH Mostly east or west facing great decliner hour line angles Tables for the great decliner (vertical), east/west gnomon adjustment Appendix 6 Sunrises and sunsets ~ for some longitudes and latitudes. Several ways of finding sunrise and sunset. Sunrise, sunset, day-length for Italian and Babylonian etc lines - many latitudes Appendix 7 Proofs and derivations ~ Geometric, trigonometric, and etc Durer geometrical model for many hour angle dials Vertical decliner hour lines, SD and SH, calendar curves, and DL Altitude/azimuth pole determination, reflecting dials Derivation of the planispheric astrolabe geometrically and trigonometrically Southern hemisphere considerations, Appendix 8 Formulae Constants and variables Spreadsheet formulae issues Actual and apparent inconsistencies in the three "classic books" on sundials Appendix 9 Templates ~ for various dial construction methods Paper cut out dials Paper popup dials Appendix 10 Bibliography ~ and cross references ~ and various guides Books, software, other references, sundial parts, sundial numerals often used Miscellaneous dates, Limiting hours by dial type, Trivia, Safety tips Testing a sundial at a location other than the design location. Summer time depiction This document is part of the www.illustratingshadows.com series of books, booklets, and software. These appendices mostly match those earlier books, however some sections have been re-organized, moved around, renumbered, and formulae standardized. Formulae involving dates use approximations thus the tables herein may disagree with other tables and sources. Please check www.illustratingshadows.com for updates to this document. This document may be copied or given to others provided credit is retained to www.illustratingshadows.com and Simon Wheaton-Smith, and no charge made. 1 APPENDIX 1 – TRIGONOMETRY AND GEOMETRY THE EQUATORIAL DIAL The equatorial dial plate, as far as hours go, the pole is independent of latitude. Latitude comes into and the equator play only for the sunset line, and setting the gnomon to latitude, with the dial plate holding the hour lines, being set to co-latitude, or 90˚– latitude. Longitude and EOT must be applied. The dial plate parallels the equator, which is why this is called an equatorial dial. Some people call the armillary dial an equatorial dial. The time cannot be read as the year approaches an equinox. The hours are marked by 15˚ radials. THE ARMILLARY DIAL The armillary dial is often called, incorrectly, the pole an equatorial dial. The word armillary relates to an “arm” bracelet surrounding the polar axis. Some equatorial dials do include an armillary at their edge, this allows the time to be read at the equinoxes. Longitude and EOT must be applied. The gnomon and the dial plate are set to latitude. The hours are marked by 15˚ radials. Some armillary dials have a number of circles added, they show the solstices, and a depiction of the ecliptic, which is the plane that the earth floats on as it orbits the sun. THE POLAR DIAL The polar dial plate parallels the polar axis, it is an armillary dial that has been flattened out. It faces the equator. Longitude and EOT must be applied. The gnomon and the dial plate are set to latitude. The hours are marked by the trigonometric “tangent” of the hour time 15˚. A polar dial can be rotated 90˚ 90 degrees around the gnomon, then the dial plate, while still paralleling the polar axis, also parallels the meridian. It is then called a meridian dial. 2 appendix 1 ~ geometry and trigonometry THE HORIZONTAL DIAL The horizontal dial has a dial plate that parallels the earth’s surface, hence it is horizontal. See the picture to the left. Longitude and EOT must be applied. The gnomon is set to latitude. The hour lines are marked by the trigonometric “tangent” of the hour time 15˚ multiplied by the trigonometric “sine” of the latitude. THE VERTICAL DIAL A vertical dial facing the equator is the same design as a horizontal dial for the co-latitude, or, 90˚–latitude. See the picture to the right. But if a longitude correction is built in, then it is in the opposite direction because the shadow moves in the opposite direction compared to a horizontal dial. It all works out well. Vertical dials may sometimes not face true south, they are called vertical decliners since the wall on which the dial is mounted declines. In the extreme, they parallel the meridian, when they become meridian dials. the sun The picture to the left is one based on the work of Albrecht Durer in 1525, and shows a horizontal dial joined to a vertical dial, and an equatorial dial meets them at their junction. The picture to the right is the same as the one above, however, the vertical dial is rotated, or declined. These provide a pictorial overview of most dial types, and from these are derived the geometrical methods, and from those, the mathematical formulae for sundial design. appendix 1 ~ geometry and trigonometry 3 Relationship between horizontal dials and vertical dials, except that as the shadows rotate in opposite directions, longitude correction if used, is reversed between horizontal and vertical dials. HORIZONTAL VERTICAL DIAL DIAL 22 23 0 1 2 3 21 2 fr sol 20 4 1 fr sol Sol 19 5 pole equator 18 6 17 7 16 8 9 15 14 13 12 11 10 10 11 12 13 14 latitude 60 latitude 30 Latitude 60 h-dial 9 10 1112 13 14 15 latitude 30 latitude 60 Latitude 30 h-dial 8 16 9 X 15 1011 1314 equator pole 1112 1415 10 x 16 9 17 4 appendix 1 ~ geometry and trigonometry Trigonometry has a set of functions that are defined that make working with angles and lines easy using simple mathematics. The pictorials at the start of this appendix allow dial types to visualized, geometric methods derived, and from those, trigonometric of pure mathematical formula to be derived. There are certain truisms about the geometry and trigonometry, or trigonometric functions. Pythagoras's Theorem sqrt(hypotenuse)= sqrt(adjacent**2+ hypotenuse opposite opposite**2) angle adjacent A right angles triangle has one angle that is 90 degrees, and the side opposite that angle is called the hypotenuse. For an angle other than the right angle, there are defined sides called the opposite side and the adjacent side.