Enhanced using opaque projection of a facsimile copy of the original letter
preserved at the University of Virginia's Alderman Library at Charlottesville.
Transcription of Mr. Clay's Letter
Dear Sir Poplar Forest Aug. 23.11
While here, & much confined to the house by my rheuma-
tism, I have amused myself with the calculating the hour lines of a
horizontal dial for the latitude of this place which I find to be 37°-22'-26"
The calculations are for every 5 minutes of time, and are always exact to
within less than half a second of a degree. As I do not know that anybody
here has taken this trouble before, I have supposed a copy would be ac-
ceptable to you. It may be a good excercise for Master Cyrus to make
you a dial by them. He will need nothing but a protractor, or a line of
chords & dividers. A dial of size, say of from 12.i. to 2.f.square, is the
cheapest & most accurate measure of time for general use, it would, I
suppose, be more common if every one possessed the proper horary
lines for his own latitude. Williamsburg being very nearly in the
parallel of Poplar Forest the calculations now sent would serve for all
the counties in the line between that place & this, for your own place,
New London, & Lynchburg in this neighborhood. Slate, as being less
affected by the sun is preferable to wood or metal, & needs but
a saw & plane to prepare it, and a knife point to mark the lines and
figures. If worth the trouble, you will of course use the paper in-
closed; if not, some of your neighbors may wish to do it & the effort
to be of some use to you will strengthen the assurances of my great es-
teem & respect
Th. Jefferson
Mr. Clay
Thomas Jefferson's Sundial Calculations
Comparison to Computer Calculation
PC TJ PC TJ
HOURS - min:deg-min-sec HOURS - min:deg-min-sec
XII / XI -00/60: 00 - 00 - 00 III / VIII -00 31 - 15 - 30
-05/55: 00 - 45 - 32 -05 32 - 22 - 43 32 - 22 - 49
-10/50: 01 - 31 - 05 -10 33 - 31 - 19
-15/45: 02 - 16 - 42 02 - 16 - 41 -15 34 - 41 - 23
-20/40: 03 - 02 - 24 -20 35 - 52 - 57
-25/35: 03 - 48 - 12 -25 36 - 06 - 04
-30/30: 04 - 34 - 09 04 - 34 - 08 -30 38 - 20 - 48
-35/25: 05 - 20 - 15 -35 39 - 37 - 12
-40/20: 06 - 06 - 33 06 - 07 - 10 -40 59 - 03 - 10
-45/15: 06 - 53 - 05 -45 42 - 15 - 14
-50/10: 07 - 39 - 52 07 - 39 - 56 -50 43 - 56 - 38
-55/05: 08 - 26 - 55 -55 45 - 00 - 34 45 - 00 - 33
I / X -00 09 - 14 - 17 09 - 05 - 19 IV / VII -00 46 - 26 - 05
-05 10 - 02 - 00 -05 47 - 53 - 34
-10 10 - 50 - 05 -10 49 - 23 - 03 49 - 27 - 03
-15 11 - 38 - 35 -15 71 - 51 - 24
-20 12 - 27 - 31 -20 52 - 28 - 07
-25 13 - 16 - 55 13 - 16 - 54 -25 54 - 03 - 45
-30 14 - 06 - 49 14 - 06 - 48 -30 55 - 41 - 28
-35 14 - 57 - 15 -35 57 - 21 - 17
-40 15 - 48 - 16 15 - 48 - 15 -40 59 - 03 - 10
-45 16 - 39 - 54 16 - 39 - 53 -45 60 - 47 - 07
-50 17 - 32 - 10 -50 62 - 33 - 05
-55 18 - 27 - 07 -55 64 - 21 - 03
II / IX -00 19 - 18 - 49 V / VI -00 66 - 10 - 56
-05 20 - 13 - 16 20 - 13 - 15 -05 68 - 02 - 41
-10 21 - 08 - 31 -10 69 - 56 - 12
-15 22 - 04 - 38 -15 71 - 51 - 24
-20 23 - 01 - 38 23 - 02 - 24 -20 73 - 48 - 08
-25 23 - 59 - 35 -25 75 - 46 - 22
-30 24 - 58 - 30 -30 77 - 45 - 46
-35 25 - 58 - 28 -35 79 - 46 - 22
-40 26 - 59 - 30 -40 81 - 47 - 54 81 - 47 - 56
-45 28 - 01 - 41 -45 83 - 50 - 14 83 - 50 - 19
-50 29 - 05 - 02 29 - 05 - 08 -50 85 - 53 - 09
-55 30 - 09 - 37 -55 87 - 56 - 29
VI / VI -00 90 - 00 - 00
Latitude 37.3739° = 37° 22' 26"
Notes About the Tables
- Only those angles calculated by TJ that do NOT agree (to the second) are given in his list.
- The angles included with his letter are symmetric relative to Noon and increase both counterclockwise to 11,...,6AM and clockwise to 1,...,6PM, the latter equal to 90°.
- The angles presented in the accompanying table are given in conventional time with the AM times inverted to read in decreasing order, e.g. 11:55 AM is equivalent to 12:05 PM, etc.
- Nine (9) times differ by 1 second of angle, all of which are larger for the computer value. Given that there are six hours of 12 values per hour for a total of 72 entries then it is likely that 7 will have the least significant digit equal 5 and half of those will round up and half round down if rounding is performed, if it is not than 3 or 4 values could be expected to be in disagreement by one second, that there are more than twice that can't be accounted statistically!
- There are five slightly larger differences: one(1) difference of 2", one (1) of 4", one (1) of 5", and two (2) of 6" of angle, all of which are smaller for the computer value.
- There are four more significantly larger differences: one (1) of 37", one (1) of 46", and one (1) of exactly 4' 0" being a difference of 27'3" and 23'3" which inspection of the facsimile confirms it mostly likely a reproducing (copy) error and not a computational one.
- Finally there is one very large difference of 8' 58" at 1:00 o'clock. Since there is approximately 9 degrees difference between both 12 and 2 this largest error would introduce only a 1/60 (9'/9°) or one minute in time error at 1:00 PM.
Construction of Jefferson's PF Dial
- Associates (PhiAssoIrontO) of the Ebony Dungeon have sandblasted a 1 foot square 3/4" thick slate tile with the Hour lines given above. We had weathered it for two years and then hesitated to offer it to Poplar Forest because it was a "re-creation." Then a December 9, 2001 news story that Monticello spent $10000 to recreate another of Jefferson's sundial designs convinced us it was worth offering ours for free!
- By removing its four corners, the Sundial becomes an octogon corresponding to the perimeter design of the villa.
- Since the angles from Noon to 8AM and 4PM measure 46° 26' 5", their intersection at the foot of the gnomen forms a 93° angle which we have used to inscribe a square (almost perfect) whose corners further divide the dial into the four hexagons reflecting the floorplan of the villa.
- Thomas Jefferson gives no instructions for the design of the shadow casting gnomen. In our design of Suntiles®, we find that thin (3/32" diam.) brass rod makes a durable, easily fabricated and stoutly mounted gnomen.
- To further individualize the sundial, strengthen the gnomen, and make the shadow interesting, a style (vertical post) has been added with a twofold (front to back, bottom to top) design of a "P" sitting atop an "F".
- It was our hope that Poplar Forest would accept a gift of the above pictured dial for display at the villa and that it will help to educate young and old on Mr. Jefferson's breadth, and his grasp and appreciation of science and mathematics. It is our pleasure to say that they accepted it on May 30, 2002.
How Accurate Will The Sundial Be?
Apparent Solar Time
- If we "set the clock" on the first days of Spring and Fall (Equinoxes) at 6AM when the sun rises and at 6 PM when the sun sets (on a clear horizon!) and every day take Noon to be when the sun is at its zenith, the time we keep is called "Apparent Solar Time." The accuracy of the Sundial then depends on how carefully we can observe the shadow.
- If it were square, the Sundial base would be 12 inches on a side. The minimum length (radius) of the shadow on the base would be 6 inches at 6 AM and 6 PM. But a practical length of the shadow would be at least 8 inches between 8 AM and 4 PM.
- At a distance of 8 inches from the foot of the shadow, it would take slightly more than a minute (67.67 seconds of time) for the shadow to move one millimeter. If we can watch the shadow move the thickness of a sheet of ordinary paper (500 sheets is about 25 millimeters (2 inches) thick then one sheet is one tenth of a millimeter) it would take just under 7 seconds.
- There are 90 degrees between 6 and 12 or 90 degrees x 60 minutes of arc per degree x 60 seconds of arc per minute of arc or 324,000 seconds of arc in 6 hours which means that there are 15 seconds of arc for each second of time.
- Thomas Jefferson calculated his lines to within one second of arc. If the lines could be drawn to his specifications they would be accurate to 1/15th of a second of time. With a shadow taking 7 seconds to move 0.1 millimeter than in 1/15th of a second it would move 0.1 millimeter per 7 seconds x 1/15 seconds or just about 1/100 of a millimeter.
- A fine human hair is 5/100ths of a millimeter, so we can conclude that this kind of precision is like splitting hairs.
Mean Solar Time
- After 1650 and the development of reliable mechanical clocks that kept constant time, every 24 hours lasted the same amount of time. Apparent Solar Time was found to run fast and slow over the year by a much as 16 minutes a day depending on the seasons.
- To smooth out these fluctations and keep agreement with mechanical clocks, Mean Solar Time was devised. To keep Mean Solar Time the sundial has to be corrected by adding or subtracting a given number of minutes (and seconds) for each given day of the year. This set of corrections is called "The Equation of Time," but it's not really an equation - it's a List which can be made into a Graph.
- The yearly list of numbers that make up the Equation of Time is actually approximate (to about one-quarter of a minute) because of leap year.
- The Sundial reads Mean Solar Time without correction (Equation of Time = 0) on April 15, June 14, September 1, and December 26.
- The Sundial runs fastest between April and June on May 14 and has to have 3min44sec subtracted, and between September and December on November 4 when16min23sec must be subtracted from its time.
- The Sundial runs slowest between December and June on February 12 when 14min20sec should be added to its time and between June and September on July 26 when 6min25sec must be added to give Mean Solar Time.
- In Jefferson's day these corrections could have been made to give a sundial accuracy to within 1/4th of a minute anywhere in Virginia (or anyplace else) and a graph could be made by connecting these points on a timeline with a smooth rounded wave curve.
Standard Time
- Had Mr. Jefferson lived another 50 years to the mid-1870's then his Dial and Mr. Clay's Dial would each have to have a different correction made to give correct "clock" time. Up until then, Mean Solar Time still measured the corrected Noon locally when the sun was at its zenith.
- Since the sun appears to travel east to west, Noon in Williamsburg at 76.75 degrees compared to Poplar Forest at 79.25 degrees west longitude had its Noon 2.5 degrees x (24 hours / 360 degrees) x (60 minutes / hour) or 10 minutes earlier.
- By the 1880's with the development of railroads and their time schedules it became necessary to adapt local time to regional time. Ultimately a clock system was adopted with 24 regions running east to west around the world between north-south lines of longitude 15 degrees apart. This time system is called Standard Time.
- For a Sundial to be corrected to Standard Time then its longitude had to be compared to the 15 degree intervals starting at the Prime Meridian at Greenwich England. For Virginia, the reference longitude was at 75 degrees. Then given Poplar Forest's longitude of 79.25 degrees or 4.25 degrees x 4 minutes / degree or 17 minutes later than Eastern Standard Time at 75 degrees. Thus when the sun is at its zenith and Noon on the Sundial on April 15 (zero equation of time correction) it will be 12:17 on the clock, but on July 26 (+6:25 equation of time correction) it would be 12:23:25.
Copyright 2002 by P. E. Field