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# Refraction (bulkeley’s glass)

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# Refraction (bulkeley’s glass)

Slides from presentation by Robert Goulding, Ph.D. (University of Notre Dame) for the benefit of the Thomas Harriot Summer Series.

Slides from presentation by Robert Goulding, Ph.D. (University of Notre Dame) for the benefit of the Thomas Harriot Summer Series.

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### Refraction (bulkeley’s glass)

1. 1. Refraction (Bulkeley’s Glass) June 23, 2021
2. 2. Measuring Refraction (1597)
3. 3. Witelo’s tables (13th century; printed 1572) 3
4. 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. 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. 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. 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. 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
9. 9. Ptolemy and Witelo Air-water tables Incident Refracted (P) (Refractio) Refracted (W) Refractio (W) 10° 8° 2° 7° 45′ 2° 5′ 20° 15° 30′ 4° 30′ 15° 30′ 4° 30′ 30° 22° 30′ 7° 30′ 22° 30′ 7° 30′ 40° 29° 11° 29° 11° 50° 35° 15° 35° 15° 60° 40° 30′ 19° 30′ 40° 30′ 19° 30′ 70° 45° 30′ 24° 30′ 45° 30′ 24° 30′ 80° 50° 30° 50° 30° 9
10. 10. Cover sheet to Harriot’s 1597 refraction investigations MS 6789, fol. 409 10
11. 11. The cross-staff (or Jacob’s staff) 11 From Gemma Frisius, De radio astronomico (1557)
12. 12. Accuracy of Harriot’s (and Witelo’s/Ptolemy’s) results θ Witelo Harriot sine-law Witelo’s error Harriot’s error 10° 7° 45′ 7° 21 1/2′ 7° 28′ 17′ 23.5′ 20° 15° 30′ 14° 46′ 14° 50′ 40′ 4′ 30° 22° 30′ 21° 42′ 22° 30′ 18′ 40° 29° 28° 44′ 28° 46′ 14′ 16′ 50° 35° 34° 57′ 35° 1′ 1′ 4′ 60° 40° 30′ 40° 28′ 40° 26′ 4′ 2′ 70° 45° 30′ 44° 41 1/2′ 44° 44′ 46′ 2.5′ 12
13. 13. On the cover: a clue in cipher “Balles of wacs bāsn & riŋ” (wax, basin, and ring) 13
14. 14. A suggested reconstruction 14
15. 15. Trying it out My angle-sighting instrument (in place of an astrolabe) and “cross staff” 15
16. 16. Trying it out “Balles of wax, basin, and ring” 16
17. 17. Trying it out Taking a sighting 17
18. 18. Trying it out Lining up “dimple” and dropping the ring 18
19. 19. Accuracy of my results θ bc ρ real ρ (SL) error 30° 3″ 22° 40′ 22° 5′ 35′ 45° 4 ½″ 32° 3′ 32° 7′ 4′ 50° 5″ 34° 49′ 35° 10′ 21′ (Depth of water in tank: 7 3/16″) 19
20. 20. Finding the sine “law” (1601)
21. 21. Calculating ray lengths MS 6789, fol. 266 21
22. 22. The table, completed cbd/bdg chd/hdg/fda ida/hdb cd bc ch hd hd 10° 13° 15′ 3° 15′ 173648 984808 4246848 4362994 757626 10° (alt) 13° 18′ 3° 18′ 173648 984808 4230298 4346886 754829 20° 26° 56′ 6° 56′ 342020 939693 1968269 2207732 755089 30° 41° 48′ 11° 48′ 500000 866025 1118439 1500302 750151 40° 58° 20′ 18° 20′ 642788 766044 616809 1174927 755229 50° 86° 40′ 36° 40′ 766044 642788 58243 1001695 767343 sinus totus: (bd) (bd) (cd) (cd) (bd) φ (incident) ρ (refracted) δ (refractio) sin φ cos φ cot φ csc ρ csc ρ.sin φ 22
23. 23. MS Add. 6789, fol. 89 The Refraction Tables 23
24. 24. Detail of air-water table sin 31°/sin 22° 41′ = 1.335 sin 34°/sin 24° 45′ = 1.335 sin 37°/sin 26° 47′ = 1.335 (et cetera!) 24
25. 25. Bulkeley’s Glass (1601?)
26. 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. 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

### Notes de l'éditeur

• 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.