4. Ptolemy’s table for refraction, air to water
Bodleian Library, MS Savile 24 (c. 1580)
10° 8°
20° 15.5°
30° 22.5°
40° 29°
50° 35°
60° 40.5°
70° 45.5°
80° 50°
4
5. Ptolemy’s table for refraction, air to glass
Bodleian Library, MS Savile 24 (c. 1580)
10° 7°
20° 13.5°
30° 19.5°
40° 25°
50° 30°
60° 34.5°
70° 38.5°
80° 42°
5
6. Bodleian Library, MS Savile 24 (c. 1580)
Ptolemy’s table for refraction, water to glass
10° 9.5°
20° 18.5°
30° 27°
40° 35°
50° 42.5°
60° 49.5°
70° 56°
80° 62°
6
7. Comparing Ptolemy and modern refraction values
Air to water measurements, according to Ptolemy and sine law of refraction
Incidence Ptolemy Sine law Error
10° 8° 7° 29′ +30′
20° 15.5° 14° 51′ +39′
30° 22.5° 22° 1′ +29′
40° 29° 28° 49′ +11′
50° 35° 35° 4′ −4′
60° 40.5° 40° 30′ 0′
70° 45.5° 44° 48′ +42′
80° 50° 47° 37′ +143′
7
8. First and second differences in air-water table
The hidden structure of Ptolemy’s refraction tables
Incidence Refraction
10° 8°
20° 15.5°
30° 22.5°
40° 29°
50° 35°
60° 40.5°
70° 45.5°
80° 50°
First differences Second differences
7.5°
.5°
7°
.5°
6.5°
.5°
6°
.5°
5.5°
.5°
5°
.5°
4.5°
Kepler, Optics (1605):
… the fault lies in Witelo’s
refractions. You will find this all
the more plausible if you pay
attention to the increments of the
increments in Witelo. For, they
grow by 30′. So it is certain that
Witelo adjusted the refractions
that he had obtained by
experiment, so as to put them in
order through the equality of the
second differences.
8
26. The Elizabethan “Telescope”
From Thomas Digges’ 1571 edition of his father Leonard’s Pantometria
(NB: First patent for a
telescope (of the
“Galilean” design)
received by Hans
Lipperhey in
Middelburg,
Netherlands, October
1608)
26
27. Harriot’s method for finding the point of burning
27
Harriot is looking for ZB (longitudo lineae concursus)
He uses a measure of refraction, and a complex series of
calculations, to find YB (linea egressionis) and the final angle
of refraction from the lens (= ∠YZB). Then:
tan ∠YZB = YB / ZB
Editor's Notes
Harriot’s results were close to Ptolemy’s (via Witelo) – which was not surprising, since they were not bad experimental values in the first place. The nature of his linear method meant that it was more prone to error at small angles (where the linear values were cramped together at the bottom of the scale), and very accurate at larger angles, where the linear values, or tangents of the angles, spread out more and more widely along the scale. And we can see the consequences here. For small angles (< 45 degrees), Harriot’s measurements are no better than Ptolemy’s and sometimes worse. For larger angles, they are often much better.
This is my hypothetical construction. [explain] [and note that, contra Lohne, it is not necessary to have an upright rule in the tank; only a horizontal one]
The “balls of wax” were likely used to mark the point in each trial that the ring touched the staff laid at the bottom of the tank.
To be honest, it was, in fact, one of several possible reconstructions I came up with. And I didn’t know which would be usable, and which (if any) would give accurate results. So the only option left was to redo the experiments.
Note the lining up of the dimple.
Conclusion: if I was able to obtain such good results with very crude instruments, it is plausible both that Harriot could have obtained much better using his staff (and with an assistant at hand!) – and that this was in fact the method that he used.
