Abstract
The content of those folios upon which the interpretation presented in this book substantially rests is presented in full and discussed exhaustively in this appendix.
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Notes
- 1.
The content of folio 90 recto has been discussed by various authors. The most comprehensive account is Damerow et al. (2004, 224–227). It bears a crowned unicorn as a watermark which it shares with almost all other folios containing consideration that can be attributed to an early phase of Galileo’s considerations regarding projectile motion.
- 2.
90 V, T1.
- 3.
See the interpretation of C02 in the discussion of 189 V in Sect. 13.20.
- 4.
90 V, T2.
- 5.
90 V, T3.
- 6.
90 V, C02.
- 7.
90 V, C01.
- 8.
The other two instances of a scalar addition are contained on folios 90 and 110. Cf. Damerow et al. (2004, 223–229 and 234). Just like on the obverse of 115, so the obverse of 90 contains notes related to the pendulum plane experiment.
- 9.
115 V, T1.
- 10.
115 V, C02.
- 11.
Galileo’s base unit 180 approach is characterized by the assumption that the radius of the fundamental circle measures 180 length units and is traversed in 180 time units. Cf. the discussion of 166 R in this appendix.
- 12.
115 V, C03.
- 13.
That the number 235 represents Galileo’s guess of the time or possibly of a lower limit of the time of motion on an arc is supported by the following argument. On folio 189 V, Galileo had rescaled his numerical result for the time of motion along the path of his best, the eight-chord approximation of the arc, calculated under the assumption that a vertical radius of 100,000 distance units would be fallen through in 100,000 time units, to yield that if the radius was instead assumed to be fallen through in 180 units, then fall along the eight-chord path would be completed in a time of 235 80∕100 of these units. Galileo knew that the actual time of motion along the arc should be slightly smaller than the value thus calculated and might thus have rounded the figure to 235. The number 235 moreover reoccurs in a calculation on folio 184 V, which receives a consistent interpretation if it is assumed that it represented Galileo’s best guess for the time of motion on the arc (cf. discussion of 184 V). Modern analysis shows that the actual value for time of motion along the arc differs only about four in a thousand from the assumed value of 235.
- 14.
115 V, D02A.
- 15.
115 V, C01.
- 16.
115 V, D01A–D01E.
- 17.
115 V, D01G.
- 18.
115 V, D01F.
- 19.
121 V, D01B and D01C.
- 20.
121 V, C01, C02, and C03. C02 has incorrectly been transcribed in the electronic representation of Galileo’s Notes on Motion.
- 21.
121 V, C04.
- 22.
121 V, C05.
- 23.
121 V, C13.
- 24.
121 V, C08.
- 25.
121 V, C06.
- 26.
121 V, C06.
- 27.
121 V, C07 and C09–C12.
- 28.
129 R D01A, D02B and D03B.
- 29.
129 R, D01A.
- 30.
129 R, D03A.
- 31.
In D02B Galileo joined the points e and c by a horizontal line. That point e lies at the same height as point c, the junction point of the broken chord, is a mere coincidence of the concrete configuration Galileo had chosen in the diagram. It may well have been the purpose of the rough sketch D01A, by choosing a more extreme position of the junction point, to show that this was in fact not an essential property of the construction.
- 32.
Compare the discussion of Galileo’s work on 174 R in Chap. 7.
- 33.
129 R, T1.
- 34.
129 R, T2.
- 35.
129 R, T3.
- 36.
129 R, T5.
- 37.
129 R, T4.
- 38.
The folios bearing this exact same watermark are 127, 130, 140, 165, 176, and 190. Interestingly, three and possibly even four of these folios are part of a double folio. 127 is connected to 126, 140 to 141, and 190 to 191. Judging from an identical fissure on both pages, 130 and 131 may also have been connected when Galileo wrote on them and then separated only later.
- 39.
130 V, 130 VD01A.
- 40.
130 V, T1A-E.
- 41.
130 V, T1F and T2.
- 42.
130 V, T2.
- 43.
130 V, T1A.
- 44.
130 V, T1A and T1B.
- 45.
130 V, T1C.
- 46.
130 V, T1D.
- 47.
130 V, T1E.
- 48.
A second elaboration of essentially the same proposition is contained on 168 R.
- 49.
130 V, T2.
- 50.
Galileo uses the letter q as a label twice. For disambiguation I refer to the second point labeled q as q′.
- 51.
That the points 4, y, 2, and 1 which mark mean proportionals are all positioned on the same line running from 4 to x initially seems not to have been clear to Galileo, as he first determined the position of the points individually by calculation and only later added the line 4x.
- 52.
Galileo added the letter θ just once on the right margin of his diagram, indicating all intersections of the perceived extension of the lower parts of the bent planes cf, ce, cb, and cn to a horizontal running through a.
- 53.
The points marking the corresponding mean proportionals on the other inclined planes are lettered 4, d, and k. The last point defined by the intersection of 4k and cn remained unlettered.
- 54.
130 V, T1F.
- 55.
130 R, C01.
- 56.
130 R, T1.
- 57.
This distance time table has not been transcribed in the electronic representation of Galileo’s Notes on Motion.
- 58.
130 R, C03.
- 59.
130 R, C04.
- 60.
I have not measured the length of the line ax or any other line in Galileo’s diagram from which this length can be inferred directly. Yet from the known width of the sheet of 210 millimeters, I infer a side length of 175 millimeters which divided by 180 gives a value of approximately 0.97 millimeters for Galileo’s punto in good accordance with the value of the units that can be inferred from other scaled diagrams. Cf. the discussion of folio 166 in this appendix.
- 61.
130 R, C042.
- 62.
148 R, D01A.
- 63.
148 R, D01D and D01E.
- 64.
148 R, T1.
- 65.
When Galileo originally drew the timeline, it started at a and ended at b. The point labels a and b were successively changed to q and p (compare Fig. 13.6). Note T1 had been formulated with respect to the original labeling. Thus also in this short text, every a was turned into a q and every b into a p, respectively. A corresponding change cannot be observed in the list of times to the left, which must thus have been produced after the labels were changed in the timeline and in the short entry.
- 66.
148 R, T1.
- 67.
148 R, C06.
- 68.
148 R, C01.
- 69.
148 R, C02.
- 70.
148 R, C04.
- 71.
148 R, C03.
- 72.
According to the description given in T1, it is the overall length of the timeline qp representing the time along ac which is initially given and with reference to which all other times are determined. Galileo, however, had conventionally decided to have the time along dc represented by qr to be measured by 120. He wrote this down in C07 first and only then determined qp noting the result next. He then crossed out the first entry and rewrote it underneath what had previously been the second line such that the order of the final list thus produced conformed to the order in which the distances had been provided in T1.
- 73.
Corresponds to: “…et ut ac ad cd ita fiat pq ad qr erit qr tempus per dc seu per bc.”
- 74.
153 R, C02.
- 75.
Corresponds to: “Sit ut cd ad do ita rq ad qs erit qs tempus per df.”
- 76.
153 R, C03.
- 77.
Corresponds to: “Fiat rursus ut ca ad av ita tempus pq ad qt erit qt tempus per ab.”
- 78.
153 R, C04.
- 79.
Galileo had initially miscalculated 3 ∗ 155 1∕3 to 366 instead of 466 and thus had arrived at an overall result of 233. That this could not be right would have been immediately obvious as the overall time pq had amounted to the smaller 219 1/3. The calculation is not transcribed correctly in the electronic representation of Galileo’s Notes on Motion.
- 80.
This conforms to the choice based on which also the numerical values for the times of motion listed in C07 and C08 were calculated, namely, to have the time along the chord dc be represented by the length of the vertical diameter dg.
- 81.
The reason why Galileo may have constructed this second timeline is suggested by the presence of some geometrical elements in the motion diagram on 148 R which Galileo had adopted from his diagram and the related considerations on double page 156, 157. These comprise the arc dcsg, the chords ds and cg, the point o marking the intersection of the latter two chords, as well as the mean proportional between the lines do and ds, which is marked but unlettered. In Fig. 13.5 I have labeled this point r′. As argued in Chap. 5, Galileo’s construction was most likely based on the assumption that the ratio of the time of fall along the broken chord to the time to fall along the chord would be given by the ratio of ds to dr′. The conformable timeline, in particular with its construction of the time of motion to fall along the broken chord, could have served to test this assumption and would, if this had been done, have demonstrated it to be unmaintainable.
- 82.
148 R, C05.
- 83.
See the discussion of the star watermarks in Chap. 10.
- 84.
On 186 V Galileo had first relocated the lower chord cb to start at a and only then constructed a mean proportional ax between ac and the relocated bc. Galileo apparently recognized that this was not necessary and that the law of fall can be applied directly. In his argument on 150 recto ck thus assumes the role that line ax had on 186 V.
- 85.
150 R T2A.
- 86.
150 R T3.
- 87.
150 R T1A.
- 88.
150 R T1B.
- 89.
150 R T1C.
- 90.
150 R T1D.
- 91.
150 R T4.
- 92.
Ibid.
- 93.
Ibid.
- 94.
150 R C04.
- 95.
- 96.
151 R, C01.
- 97.
The sesquialterum variant of the double distance rule states that if the accelerated motion of fall over an inclined plane, here plane xy, is diverted into a uniform horizontal motion over a distance yz equal in length to the inclined plane, the overall time for the motion will be one and a half (sesqualiter) times that of the motion of fall on the inclined plane alone. See the discussion in Chap. 10. See also Damerow et al. (2004, 175–179).
- 98.
151 R, T1A.
- 99.
151 R, T1B.
- 100.
In the diagrams on 151 V and on 185 V, Galileo drew three as opposed to just one circle that he had drawn in the corresponding diagram in the Discorsi. This indeed can be seen as corroborating the assumption that Galileo is elaborating lemma three, as the different circles exemplify the universal validity of the lemma for any position of point I on the arc AC. It may in fact have been the construction of internally touching circles thus engendered that have triggered Galileo’s consideration regarding the iso-temporal surface of naturally accelerated motion and thus his sketch of the second, overlapping diagram.
- 101.
Folio 152 documents Galileo’s insight that the Sarpi letter principle was erroneous. See Damerow et al. (2004, 184–188).
- 102.
153 R, D01A.
- 103.
153 R, D02A.
- 104.
158 is the length of line scx on folio 174 R, yet no particular reason is obvious from the considerations on 174 R why Galileo should have divided this length by 4.
- 105.
153 R, C05.
- 106.
153 R, C06.
- 107.
153 R, C08.
- 108.
153 R, C07.
- 109.
The summing up of numbers in the second calculation C02 is reminiscent of Galileo’s summation of times in the case of the pendulum plane experiment.
- 110.
154 R, C01.
- 111.
Ibid.
- 112.
154 R, C02.
- 113.
154 R, C04.
- 114.
154 R, C05.
- 115.
154 R, C03.
- 116.
154 R, C08.
- 117.
154 R, C06.
- 118.
154 R, C07.
- 119.
To solve the problem, a point a′ is constructed on the inclined plane such that ba′ = ab. Then one needs to find a point X such that xa′ is the mean proportional between xb and xX. The way I have realized this in my reconstruction is by constructing a right-angled triangle in which b is the foot-point of the height with the side above xb being equal in length to xa′. Then the hypotenuse is xX. There are certainly many other ways to realize the same.
- 120.
See Damerow et al. (2001), in particular 65–82.
- 121.
157 V, D01A, D03A and D04A.
- 122.
157 V, T2.
- 123.
156 R, C01. 184776, the length of as can directly be read from a table of sines.
- 124.
156 R, C01.
- 125.
In the electronic representation of Galileo’s Notes on Motion, “media inter ga au” is transcribed with the content of C01.
- 126.
156 R, C02.
- 127.
157 V, C01.
- 128.
156 R, C03.
- 129.
157 V D03A and D04A.
- 130.
See Drake (1979).
- 131.
That the construction on 166 R is drawn to scale has already been assumed by Drake (1978, 88). He gives a value of 29/30 mm, corresponding precisely to my own measurements. From measurements on a proportional compass by Galileo, Naylor (1976) inferred a value of 0.95 mm for the smallest unit. More recently Vergara Caffarelli (2009, 169) has claimed that Galileo’s puntus is synonymous to the piccolo which is 1/240 of a Florentine braccio and should thus measure about 2.4 mm. I find his arguments not convincing. In particular it would result in very improbable dimensions for different experimental setups used by Galileo. Such, for instance, the 828 punti Galileo gives on 116 V as the height of a table would amount to a table of almost 2 meters height instead of the even today still standard height of somewhat less than 80 centimeters which results from the conventional assumption that the punto measures about 0.96 millimeters.
- 132.
166 R, T2.
- 133.
166 R, T3A.
- 134.
166 R, C02.
- 135.
166 R, T3B.
- 136.
On 192 R Galileo did calculate the time egc from rest at point e (see the discussion of folio 192 R in this appendix) yet for the base 100,000 approach. From the time of 66,326 given for motion along egc in the entry on 166 R, it can be inferred that Galileo’s calculations on which this entry was based had proceeded from the assumption that t(ec) = ec = 76, 536, thus implicitly defining a time unit not compliant with that of the base 100,000 approach.
- 137.
166 R, T3B.
- 138.
From the time of motion along 8c assumed to be 19,598 it is clear that these calculations must have been based on assuming by distance time coordination that the time along 8c is measured by 19,598, i.e., the length of 8c in the length units of the base 100,000 approach. With this choice Galileo implicitly introduced yet another unit of time measurement.
- 139.
166 R, T1D.
- 140.
166 R, T1A.
- 141.
166 R, T1B.
- 142.
166 R, T1C. The calculation is not preserved. According to the law of fall, the time is simply given by the mean proportional between the radius and the length of the circumference of the quadrant.
- 143.
166 R, T4.
- 144.
167 V, C01.
- 145.
Ibid.
- 146.
Ibid.
- 147.
Ibid.
- 148.
For a list of the folios sharing this watermark see discussion of folio 130 in this appendix.
- 149.
176 R, D02A.
- 150.
176 R, T1.
- 151.
176 R, C01.
- 152.
176 R, C01.
- 153.
176 R, C02.
- 154.
Galileo explicitly noted a time of 132 for motion on the broken chord on 189 R, C08, but the result he had calculated in C06, and which he could furthermore read off his list of results on 183 R, was somewhat greater, namely, slightly more than 132 1∕2. Why Galileo changed the value used for the root of 100,000 to 304 remains unclear.
- 155.
176 R, C03.
- 156.
See Drake (1979).
- 157.
The folios 167 and 166 form a double folio. As the watermark of folio 166 is the same as that on folio 184, it is suggested that folios 183 and 184 were connected as a double folio as well when Galileo wrote on them. This may indeed be the reason why of the pages and pages which Galileo must have filled with the calculations that were required to produce the list on 183 R and the comparable list on 166 R, only those calculations written down on 184 have survived. It seems that once the results had been summarized in the lists, Galileo discarded the folios containing the calculations except for 184 as this folio was presumably then still attached to the page comprising the list of results.
- 158.
Should be 91,018.
- 159.
Should be 41,576.
- 160.
Should be 63,073.
- 161.
Should be 26,733.
- 162.
Should be 26,752.
- 163.
Should be 19,895.
- 164.
Should be 44,383.
- 165.
Should be 131,416.
- 166.
Should be 12,358.
- 167.
Should be 11,184.
- 168.
Should be 10,466.
- 169.
Should be 10,045.
- 170.
Should be 9,849.
- 171.
183 R, T1.
- 172.
183 R, C04.
- 173.
183 R, C05.
- 174.
183 R, C06.
- 175.
Ibid.
- 176.
183 R, C07.
- 177.
183 R, C08.
- 178.
183 R, C09.
- 179.
183 R, C01, incorrectly transcribed in electronic representation of Galileo’s Notes on Motion.
- 180.
184 V, T1.
- 181.
184 V, C01.
- 182.
184 V, C02.
- 183.
184 V, C05.
- 184.
Ibid.
- 185.
184 V, C09.
- 186.
184 V, C07.
- 187.
184 V, C09.
- 188.
184 V, C20.
- 189.
184 V, C15.
- 190.
184 V, C12.
- 191.
184 V, C14.
- 192.
184 V, C13.
- 193.
184 V, C16.
- 194.
184 V, C19.
- 195.
184 V, C21.
- 196.
184 V, C08 and C21.
- 197.
184 V, C11.
- 198.
184 V, C17.
- 199.
184 V, C06.
- 200.
184 V, C17.
- 201.
Erroneous result noted, see discussion of C03 above. The list is completed by using the recalculated, correct value for t(xr).
- 202.
184 V, C17. Minutum is the name given to the time unit in which the calculations are carried out on 166 R.
- 203.
184 V, C05.
- 204.
184 V, C10.
- 205.
185 R, C01.
- 206.
This line has not been transcribed in the electronic representation of Galileo’s Notes on Motion.
- 207.
185 R, C02.
- 208.
185 R, T1.
- 209.
The only other folio of the Notes on Motion bearing the exact same watermark is folio 192.
- 210.
186 V, T1A.
- 211.
Ibid.
- 212.
186 V, T1B.
- 213.
Ibid.
- 214.
186 V, T2.
- 215.
189 V, D01B.
- 216.
189 V, C01.
- 217.
As argued in Chap. 4 Galileo most likely used a water clock for measuring intervals of time in the experiment. If the flow rate is constant over the time of measurement, the weight of the water that has flowed out during measurement can be used as a measure of the time elapsed. The smaller quantities which are being added up could thus be the weights placed on a balance to weigh the water or else the volume of containers filled with the water that had run out.
- 218.
189 V,T3A.
- 219.
After this correction Galileo used ab consistently to refer to the diameter instead of the radius.
- 220.
Galileo had apparently counted full periods, i.e., the number of times the pendulum bob had returned to his initial position during the measurement. His corrections seem to be due to the realization that dividing the time measure by two times the number of periods counted had given half the pendulum period of the pendulum and not the quarter period he was interested in for his successive theoretical investigations, i.e., the time it took the pendulum to move from one of the turning points of motion to the lowest, the equilibrium position of the bob.
- 221.
189 V,C02A.
- 222.
The length of a chord subtending an arc of a given angle is twice the sine of half that angle in the same circle. I have not been able to identify which sine table Galileo used concretely. The value Regiomontanus gives for the sines of 8 degrees for a circle of radius 10,000,000 is 139,173. See Peuerbach and Regiomontanus (1541, E3). Hill (1994) assumes that the inclined plane was not 6700 but twice as long, namely, 13400 punti, and was inclined at an angle of about 16 degrees and 10 minutes. This is based on the not further justified surmise that the number 27,834 represents the length of half the chord. Looking up in Copernicus’ table of sines (Kopernikus 1542), the value closest to 27,834 Hill read off an angle of 16 degrees and 10 minutes. Whereas a chord length 27,834 can directly be read off from the sine tables, Hill’s alleged chord length of twice that value does not correspond to any of the angles listed in steps of 1 minute. Moreover, Hill’s interpretation leads together with the assumption that the unit in which Galileo gave the lengths corresponds to about 0.96 mm to the rather implausible conjecture that Galileo used an almost 13-meter-long inclined plane, which at an inclination of 16 degrees must have been elevated more than 3 1/2 meters at one end.
- 223.
189 V, T3B. Galileo had originally written 48142, which he corrected to 48143 most likely to account for fractions.
- 224.
189 V,C03. As Hill has already remarked, the actual result ought to be 13877, but Galileo had failed to carry a one when calculating the product.
- 225.
189 V,C06.
- 226.
189 V,D03A.
- 227.
189 V, T1.
- 228.
189 V, C04. None of the calculations to establish the values in the table can be identified. In view of the simplicity of the example, Galileo could easily have done the required calculations in his head. The values testify that the law of fall and length time proportionality were resorted to in constructing the example. The first entry in the table, the length of the distance ae, has been corrected from a now illegible value to 9. As numbers underneath the distance-time table suggest that Galileo had originally chosen ae to measure 10 units, but realized that no smaller natural number could be found such that the mean proportional between 10 and this number, representing the time of motion along ad, would likewise be a natural number.
- 229.
189 V, C12.
- 230.
189 V, C11.
- 231.
189 V, C13.
- 232.
189 V, T4.
- 233.
189 V, T2.
- 234.
189 V, C08.
- 235.
189 V, C09.
- 236.
189 V, C09.
- 237.
189 V, C07.
- 238.
189 R, C04.
- 239.
189 R, C02.
- 240.
Wrong in electronic representation of Notes on Motion.
- 241.
Wrong in electronic representation of Notes on Motion.
- 242.
189 R, C05.
- 243.
189 R, C06.
- 244.
Incorrectly transcribed in ERGNM.
- 245.
Incorrectly transcribed in ERGNM.
- 246.
189 R, C08.
- 247.
189 R, C07.
- 248.
(D01A, D02A, D02C, D03A).
- 249.
189 R, C01. These numbers are incorrectly rendered as 7071 and 7077 in the electronic representation of Galileo’s Notes on Motion.
- 250.
These two numbers are not transcribed in the electronic representation of Galileo’s Notes on Motion.
- 251.
192 R, C03.
- 252.
192 R, C04. Galileo carried out the calculation twice because of a slight error in the first attempt.
- 253.
192 R, C03.
- 254.
192 R, T3.
- 255.
192 R, C06.
- 256.
192 R, C07.
- 257.
192 R, C05.
- 258.
192 R, C10.
- 259.
192 R, C09.
- 260.
192 R, C09.
- 261.
192 R, C01.
- 262.
192 R, C01.
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Büttner, J. (2019). Appendix: Folio Pages. In: Swinging and Rolling. Boston Studies in the Philosophy and History of Science, vol 335. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1594-0_13
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