Annotated Translation of Collectio IV

Sefrin-Weis, Heike

doi:10.1007/978-1-84996-005-2_2

Annotated Translation of Collectio IV

Heike Sefrin-Weis³

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Notes

1.
The text of Prop. 1 shows a number of idiosyncrasies in labeling parallelograms. I have followed Hu in standardizing.
2.
By construction.
3.
I, 29.
4.
I, 35.
5.
I, 35.
6.
Reference to I, 47; Hu 178, 13 + app. Hu notes that a later manuscript has added a phrase that establishes a connection to VI, 31, which is not envisaged in Prop. 1; see the apparatus to the Greek text and the commentary.
7.
For the definition of a “Minor” see X, 76, for its classification see X, 82, and for its construction (used here in Prop. 2) see X, 94.
8.
For the meaning of “rational” see the commentary.
9.
The manuscript A has “oὒτῶϛ,” Hultsch changes to the standard õτƖ (cf. also Co p. 58 A). Treweek follows him. The use of the differing conjunction is an idiosyncrasy of Props. 2–6. In the Greek text, I have kept the reading of A in all instances.
10.
For the meaning of the term “irrational” in this context see the commentary.
11.
III, 31.
12.
VI, 4.
13.
I, 47: CD² = 3ZB², and ZC² = 4ZB².
14.
VI, 23.
15.
ZC = 2ZB, ZE = ZB.
16.
Choose T on ZB, Z − B − T, with TB = 1/4ZB.
17.
ZT = 3BT; ZC = 2ZB = 8BT; TC = 5BT.
18.
V, 23 with V, 16.
19.
ZT²:(ZT² − ZH²) = 25:13; X, 9 with X, 5/6.
20.
ZT = 3BT; AB = 2ZB = 8BT; X, 9.
21.
X, 73; X, 84 a 4. The Apotome is introduced in X, 73, divided into subtypes in X, 84, with geometrical constructions in X, 85–90.
22.
X, 94.
23.
This will be shown below.
24.
II, 12.
25.
II, 13 generalized. The proof and theorem of II, 13 in the Elements specifies acute-angled triangles, but it can easily be extended (cf. Heath 1926 I, pp. 406–409). Within the present translation and commentary, I will refer to “II,12/13 generalized,” assuming that Pappus expects familiarity with the generalized versions. A theorem much like II, 12/13 generalized seems to be invoked inter aliain Prop. 7 and Prop. 8. The following piece of text is bracketed by Hultsch as a later addition: Therefore, proportion holds. As the square over CE is to the (sum of the) squares over ET and TC, together with two times the rectangle, so is the sum of the squares over ET and TZ to the square over EZ, together with two times the rectangle ZTH (in A:ZHΘ). And as one to one, so are all (to all, add. Hu). And the square over CE is equal to the sum of the squares over ET and TC, plus two times the rectangle CTH (in A:ΓEΘH).
26.
I.e., 25 BH². BH² appears as a unit of measure. The areas of squares are directly identified with numbers. This is unusual.
27.
Note the connection to Prop. 2. There the arc between point of touch and base was bisected, here it is the angle between the tangent and the base.
28.
The Binomial is introduced as a sum in X, 36, shown to be uniquely determined this way in X, 42, split up into six types in X, 47–53, and geometrically constructed and characterized in X, 54.
29.
The Line that produces with a rational area a medial whole is introduced as a difference of lines in X, 77, the uniqueness of this determination is proved in X, 83, and the line is constructed and characterized geometrically in X, 95.
30.
B on the semicircle, because ∠HBD = π/2.
31.
I.e., of a regular hexagon inscribed in the circle with diameter HD.
32.
See Prop. 2 for this intermediate step.
33.
X, 9.
34.
X, 9.
35.
X, 47 a 1.
36.
X, 54.
37.
VI, 8.
38.
VI, 4; VI, 17.
39.
Equilateral triangle HBC.
40.
X, 9.
41.
X, 84 a 5.
42.
VI, 4; VI, 17.
43.
VI, 4; VI, 17.
44.
V, 16; rectangle DHT:rectangle DHL = BH²:KH².
45.
The following phrase was bracketed by Hultsch; it translates to: For the height is equal.
46.
V, 17.
47.
X, 95.
48.
DZ secant to the circle, chosen at liberty between DC and DA.
49.
Analysis - assumption: EK = EL.
50.
ΔHKE ~ ΔHMX; ΔHEL ~ ΔHXT; VI, 4 and V, 16; KE = EL by assumption in the analysis.
51.
III, 3.
52.
oτωϛ ἅρα; the occurrence of this phrase is a peculiarity of the analysis in Prop. 4. It will be translated as “thus.”
53.
I, 29.
54.
III, 21.
55.
Converse of III, 21; see the commentary.
56.
III, 21; I, 29.
57.
∠AND = ZAED. The analysis proper ends here.
58.
III, 31. Note that this observation constitutes the resolutio for the analysis in Prop. 4. The position of E, N, A, and D on a circle is independent from the analysis-assumption.
59.
Technical term: συνθετήσεταƖ. The synthesis begins here.
60.
III, 31; compare the above resolutio. See the commentary.
61.
By construction, I, 29.
62.
See the commentary. An appeal to the converse of III, 21 is not permissible in the synthesis.
63.
III, 21.
64.
III, 3.
65.
V, 16 yields MX:TX = EK:LE.
66.
ΔMZE ~ ΔCZH, VI, 2.
67.
III, 18.
68.
I, 4.
69.
Triangle LAC is isosceles; therefore, ∠LAC = ∠LCA (I, 5); therefore, ∠CAD = ∠ACD, and triangle ACD is isosceles (I, 6). Co refers to III, 36, with corollaries; cf. 191, * Hu.
70.
AC || EM; therefore, ΔEDM is isosceles (I, 29; I, 6). Co refers to VI, 4; Hu’s Latin paraphrase suggests using VI, 4, also.
71.
I, 4.
72.
ΔEHL ≅ ΔZHL; therefore, the neighboring angles at H are equal.
73.
III, 3.
74.
III, 18; III, 3.
75.
III, 31.
76.
III, 21.
77.
III, 31; compare above.
78.
Isosceles triangle ACL; I, 5.
79.
III, 21.
80.
Isosceles triangle ELZ; I, 5.
81.
III, 3.
82.
For information on the technical term “given” (Latin:datum, Greek:δoθέν) see the introduction to Prop. 7 in the commentary. Determining givens (data) is the central task of the resolutio stage of Greek geometrical analysis (see introduction to Props. 4–12). The terminology will also be employed in Props. 28 and 29, 35–41, 42–44, and 31–34. In the latter cases, Pappus is operating outside the scope of plane geometry, and the analysis serves very different functions.
83.
I.e., the circumscribed circle, touching each of the three given ones.
84.
E is on AC. Its position must be A — E — C because of the right angle at B. H is on DC. Its position D — H — C implies a special configuration for Prop. 7. See the commentary. Z is taken to lie on DC and BE.
85.
I.e.: it is given in size. Elem. VI, 8, Porisma, and VI, 17: BC² = AC × EC. Data 52: with BC given,BC² is given.
86.
I, 47: AC² = AB² + BC², Data 52: AB, BC given ⇒ AB², BC² given; Data 3: AB² + BC² given,i.e., AC² given; Data 55: AC given.
87.
Data 57: AC × EC given, AC given ⇒ EC given. Data 4: AC, EC given ⇒ AC − EC, i.e., AE given. I, 47: BC² = BE² + EC², i.e.: BC² − EC² = BE²; Data 52: BC, EC given ⇒ BC², EC² given. Data 4: BC² − EC² given; Data 55: BE given.
88.
According to Hultsch, no such lemma is still extant. He provides a proof for “CD − HD is given” at Hu p. 193, # 4. II, 12/13 generalized: AC² = AD² + DC² − 2DC × DH; therefore: AC² − AD² = DC × (DC − 2DH). Data 52: AC², AD² given; Data 4: AC² − AD² given, i.e., DC × (DC − 2DH) given; Data 57: DC, DC × (DC − 2DH) given ⇒ DC − 2DH given; Data 4: DC, DC − 2DH given⇒ 2DH given; Data 2: DH given. Data 4: DC, DH given ⇒ HC given. I, 47: AD² = AH² + DH², i.e., AH² = AD² − DH²; Data 52: AH², DH² given; Data 4: AD² − DH² given, i.e., AH² given; Data 55: AH given.
89.
VI, 4, V, 16.
90.
Data 1 (HC and CE are given).
91.
Data 2 (AC, AH, and HC:CE are given).
92.
Data 3 (ZB = ZE + BE).
93.
H is assumed to lie on DC, with D — Z — T — C. This means that, again, only one of several possible sub-cases is discussed here; see the commentary on the purpose of Prop. 7.
94.
ΔZBT ~ ΔZEC implies BZ:ZT = CZ:ZE; Data 2: BZ, BZ:ZT given implies ZT given. Data 4: CZ − ZT = TC given. I, 47: BT² = BZ² − ZT². Data 52, Data 4: BT² given; Data 55: BT given.
95.
Data 4: DC, CT given implies DT given. BT was shown to be given already.
96.
I, 47: BT² + DT² = BD²; Data 52: BT², DT² given; Data 3: BD² given; Data 55: BD given.
97.
E is on AC, and Z is on BC. Again, the position A — E — C constitutes one of several possible cases. Furthermore, the argument will assume B— Z — H — C for the relative position of the inter section points of DE and BC, and the perpendicular. As remarked above, Pappus covers only two of a number of possible cases. The proof is completely analogous in all cases (cf. 195, * Hu). See the commentary on Prop. 7.
98.
As in Prop. 7a: I, 47: AC² = AB² + BC²; with AB, BC given, Data 52, Data 4: AC² given; Data55: AC given.
99.
As in 7a, for DH, HC, AH; II, 12/13, generalized: AC² = AD² + DC² − 2CE × AC, i.e., AC(AC + 2CE) = AD² + DC²; with AD, DC given, Data 52, 3: AC (AC + 2CE) given; AC, AC(AC + 2EC) given⇒ AC + 2EC given (Data 57); Data 4, Data 2: EC given. With Data 4: AE given.
100.
VI, 4.
101.
Data 1: CB:BA given, i.e., CE:EZ given. Data 2: CE, CE:CZ given ⇒ EZ given.
102.
I, 47: DE² = DC² − EC²; Data 52, 4: DE² given; Data 55: DE given.
103.
Data 3 (DZ = DE + EZ, and DE, EZ are given).
104.
Apply VI, 4, for similar triangles ABC, CEZ:AC: BC = CZ:EZ; Data 1: AC:BC (thus: CZ:EZ) given; Data 2: CZ given (EZ, CZ:EZ are), Data 4: BZ given (BZ = BC − ZC).
105.
H on BC; the relative position B — Z — H — C constitutes one of several possible cases. See the commentary.
106.
II, 12/13, generalized: ZC² = DZ² + DC² - 2DC × ZH; Data 52, 4: DC × (DC - 2ZH) given; Data 57: DC − 2ZH given Data 4, 2: ZH given; Data 4: ZC − ZH, i.e., HC given. Compare 7a for ΔABC with lines DH, HC.
107.
Data 4: BH given (BC - HC), I, 47: ZH² + HD² = DZ², i.e., HD² = DZ² − ZH². Data 52: DC², HC² given, Data 4, HD² (= DC² − HC²) given; Data 55: HD given; compare the argument for AH in 7a.
108.
I, 47, BD² = BH² + DH²; with BH, HD given: Data 52, Data 3: BD² given; Data 55: BD given. Compare the last step in 7a.
109.
Co p. 66, B points out that the argument in Prop. 8 implies that CA, CB are given in position and size.
110.
Prop. 8 gives the resolutio for a special configuration in one of the cases treated in Apollonius, Tangencies. The Tangencies are lost, but Pappus’ commentary on it can be found in Coll. VII. Coll. VII, Props. 102–107, are directly relevant for Prop. 8, as can be seen in the footnotes below. It is quite possible that Prop. 8 is in fact a (so far overlooked) testimony for a fragment from Apollonius’ lost work. The connection of Prop. 8 to Apollonius’ Tangencies, specifically the case of three touching circles, is noted also in Heath (1921, II, pp. 182–184).
111.
Extension of the configuration. Inconsistencies of labeling occur throughout Prop. 8 (compare 194/196 + app. Hu). Some of them probably go back to Pappus. For Prop. 8 shows clear signs of a not quite complete revision of a source text, after the insertion of additional material. See the commentary on this issue.
112.
For a proof, compare Coll. VII, Prop. 102, p. 826 Hu. (Jones 1986a, Vol. 1 p. 234, # 164). The proposition in Coll. VII is Pappus’ commentary on Apollonius, Tangencies, I, 16. Also, compare Hultsch, p. 197, #2; Co p. 66/67, Lemma in E for a different explanation via similar arcs.
113.
For a proof, compare Coll. VII, 106, p. 833/834 Hu (Jones 1986a, Vol. 1. p. 238, # 169). The proposition in Coll. VII is a lemma by Pappus on Apollonius, Tangencies I, 17. Also, compare p. 197, #3 Hu and Co p. 67, G for a different explanation via similar arcs.
114.
III, 14.
115.
The explicit argument for BM = MA, LM = MS is much more elementary than the rest of the inferences in Prop. 8. Hultsch (196, 9–16 app.) suspects interpolation. Another possibility is that Pappus himself inserted this elementary material and has not fully integrated his resulting overall argument. There are further problems with the transmitted text and its line of reasoning (see 196, 17–198, 18 + app. Hu).
116.
Hultsch deletes the following here: “and in the same way both ZH and DE and BL and LS” (196, 18/19 + app. Hu). The phrase does not fit the context of the argument as given. Perhaps it is a leftover from a version of the text that was replaced by the suspected lines discussing BM, MA, LM, LS. Compare the preceding footnote. The implicit argument given for the status of ML, LB, MS, SA as givens — which the reader is perhaps meant to supply — shows strong affinities to Prop. 7. Compare p. 197/199, #4 Hu, including a reference to notes #2 and #3 on Prop. 7. A shorter route, avoiding the connection with Prop. 7, would have been to infer B, A given ⇒AB given(Data 26) ⇒ BM, MA given (Data 7).
117.
Data 26.
118.
Data 39. Indeed, the triangle is then given in position and size as well. The ensuing argu ment does not take advantage of these facts, and this may be yet another sign that Pappus has introduced material (from the Data, this time) into an argument that perhaps did not use the Data.
119.
Appeal to Prop. 7; compare p. 193, #3 Hu. Hultsch brackets the reference to the drawing of a perpendicular (thus, the reference to Prop. 7 is eliminated) and offers alternative arguments for “CM given” at 199, # 5 Hu. Evidently, Hultsch viewed the reference to Prop. 7 as something that is not of one piece with the main body of the argument in Prop. 8. We have yet another indication for Pappus’ introduction and incomplete integration of material into Prop. 8.
120.
Data, def. 5: AR given; Then NR (its double) is given, also. The argument will use AR.
121.
Data 4; compare 199, # 6 Hu (covering NR, AR, MR, MN).
122.
III, 35; III, 36.
123.
Data 57. Note that CP is in fact given in position and size.
124.
That Z (and therefore all sides of the triangle CZP) is given, can be shown by Coll. VII, 105, pp. 830–831 Hu (Jones 1986a, Vol. 1, p. 236, # 168). That it is the point of touch for the sought circle with the circle around A, can be derived via Coll. VII, 104, pp. 828–829 Hu (Jones 1986a, Vol. 1, p. 234, # 166). Both lemmata are taken from Pappus’ commentary on Apollonius, Tangencies I, 16. The latter lemma is the converse of Coll. VII, 102, quoted above.
125.
AD > CD > DB. A more literal translation of the sentence is: (let) that by which CD exceeds DB be equal to that by which AD exceeds CD.
126.
The text has “excess” (ὑ περoχή). In Prop. 10, Pappus will use the word “difference” (δ0196;αϕoρἅ).
127.
d = AD - CD = CD − DB, and d is given.
128.
E lies on AD, A — E — D; Z lies on DB, D — B — Z.
129.
ED = AD − d, DZ = DB + d = DB + (AD − CD) = AD − d. DC = AD − d.
130.
Prop. 8: the diameter of the circle CEZ is given. Then its radius is given, also.
131.
BZ = d.
132.
Data 4.
133.
AD = DZ + BZ; Data 3; DC = DZ.
134.
Prop. 10 (in a much more general version) was announced before Prop. 7. Prop. 10 is essentially the resolutio of an analysis for a single very specific case out of several possible cases for the Apollonian problem. Construction and apodeixis are not offered. See the commentary.
135.
Whereas A labels the center of the (sought for) comprising circle with “H” here, the accompanying diagram, and parts of the text further down take the center to be N.
136.
D, E, Z will be the points of touch with the sought circle: III, 11 and 12.
137.
Here, the word used in A is δƖαϕoρἅ, whereas in Prop. 9, the word ύπϵρoχή was used.
138.
Pappus appeals to Prop. 9. However, in Prop. 9, the additional assumption was made that d = AD − DC = DC − CB, and this is not stated in Prop. 10. Pappus would have had to furnish an extension of Prop. 9, or else formulate an appropriate restriction on the configuration for Prop. 10. Hultsch p. 201, #3, supplies part of an argument, via Prop. 9, to establish that AH is given. On the issue of the gap in Prop. 10 see also appendix Hu p. 1227, and the commentary.
139.
The issue is not picked up again in Coll. IV. Perhaps Pappus intended to revise Props. 7–10.
140.
D is chosen on the circumference so that CD = BC − AD.
141.
I have translated the text as read/reconstructed by Treweek, treating the phrase “πὶ τὴν AΓ” as an explanatory addition (cf. 202, 3 + app. Hu).
142.
H on AC, A — E — H, AE = EH; T on BC, B — T — C, BT = BA.
143.
This passage contains the extension of the configuration (five auxiliary lines and points). From here, the symperasma can be directly deduced using the resulting triangles. See the commentary for a conjecture on how this might be indicative for the purpose of Prop. 11 within a group of propositions on analysis-synthesis.
144.
ΔADC ~ ΔDZC; VI, 8, VI, 4, VI, 17.
145.
ΔATC ~ ΔTZC; VI, 17, VI, 6.
146.
ΔABC ~ΔAEB: VI, 8; VI, 4, VI, 17.
147.
AH = 2AE by construction.
148.
I, 47.
149.
ΔATC ~ ΔATH (VI, 17, VI, 6), and ΔATH ~ ΔTZC has been shown.
150.
Complementary angles; DHTZ is therefore isosceles (I, 6), and TZ = TH. In the manuscript A, TH = TZ is claimed directly (202, 19 f. + app. Hu). Perhaps the manuscript reading would have been preferable.
151.
TK is perpendicular to AC by construction.
152.
I, 26 for ΔTKH, ΔTKZ.
153.
Circle with diameter AT; III, 31.
154.
III, 21.
155.
ΔABT is isosceles, and the angle at B is a right angle by construction.
156.
ΔBEK has a right angle at E, half a right angle at K; it is isosceles (I, 6).
157.
Note the similarity of the starting configuration to the one in Prop. 11. An implicit assumption in Prop. 12 is arc ABC < arc of quadrant.
158.
Choose T on AC, A — D — H — T — Z, with DT:TZ = AH:HD.
159.
III, 3.
160.
III, 31.
161.
I, 29.
162.
Prop. 12 contains several series of phrases starting with “therefore, that X”, all dependent on some single “one sought to show that Y”. Within Coll. I V, this stylistic feature is unique. To facilitate reading, I have added the implicit phrases in brackets.
163.
I, 29. The analysis in Prop. 12 is predominantly reductive and deductive (with minimal input by extension of configuration). All steps are also convertible, and the synthesis will therefore mirror the analysis exactly. See the commentary on Prop. 12, and the introduction to Props. 4–12 on analysis-synthesis for this feature of the analysis in Prop. 12 in the context of plane geometry.
164.
Above, the claim in Prop. 12 was reduced to this statement.
165.
If KD is parallel to ET, ΔDKH ~ ΔETZ (ΔKLH ~ ΔEDZ by construction; I, 29); then the above mentioned proportions hold. The claim of the statement has been reduced to yet another condition that must be fulfilled.
166.
V, 22.
167.
V, 17. At this point, the initial claim has been reduced to: LD:HD = DT:ZT.
168.
Hypothesis of Prop. 12.
169.
Using the result of the first sequence of reductions.
170.
V, 9.
171.
LA = LD − AD; DH = AH − AD.
172.
Hypothesis of Prop. 12.
173.
Since AB = BD, the reduced claim LA = DH implies that DLAB ~ DHDB must hold (I, 4).
174.
BH = AH: radii of initial semicircle; AH = LD needs to be shown (see above); therefore, the claim of Prop. 12 has been reduced to BL (= BH = AH) = LD.
175.
Beginning of the resolutio. BL = LD holds independently of the analysis-assumption.
176.
By construction.
177.
I, 29, I, 5, I, 29.
178.
BK = KD by construction; I, 4 for ΔLBK, ΔLDK. The resolutio ends here.
179.
Greek word: aἀκoλoὸθωϛ (translation: following step by step). This term was subject to consid erable debate in the discussion about the interpretation of Greek geometrical analysis and its logical structure. Some authors hold that it must mean “logically derived”, and maintain that analysis is deductive, since it proceeds “akolouthos.” I agree with Hintikka and others that it does not have to be interpreted so narrowly, and that it rather means “follows in sequence, in an orderly fashion”. Co p. 70 translates “compositio vero resolutioni congruens erit.” See the excur sus on analysis-synthesis in the introduction to Props. 4–12 in the commentary. The synthesis is not a direct logical deduction from the analysis. Further occurrences of this word and its deriva tives in Coll. IV, where regularly it does not carry the force of “logical derivation” will be noted ad locum.
180.
III, 3; III, 31.
181.
I, 5.
182.
I, 29.
183.
III, 31; III, 3.
184.
I, 4.
185.
Isosceles triangles, I, 5.
186.
Isosceles triangle ABD, I, 6.
187.
I, 26.
188.
Add AD.
189.
Hypothesis of Prop. 12.
190.
V, 18.
191.
ΔLKD ~ ΔDEZ; VI, 4.
192.
V, 22.
193.
I, 29.
194.
VI, 6.
195.
I, 27; the corresponding step in the analysis (converse) rests on I, 29.
196.
I, 29; I, 6.
197.
The meaning of the term “arbelos” is not quite clear. One of the possible meanings is “shoemaker’s knife”. Apparently, ancient shoemakers used a tool with a shape that was similar to the one formed in the figure.
198.
Only the first of the inscribed circles touches all three initial semicircles; all others touch two of the semicircles and their own predecessor and successor. Note the motivic connection to Props. 7–10. Each inscribed circle in the arbelos sequence is a solution to the Apollonian problem. Only the starting configuration, however, is directly related to the special case treated in Prop. 10; cf. Jones (1986a, p. 539). See also Hofmann (1990) II, pp. 146–164, and the notes in the commentary on Props. 13–18.
199.
τἅλαμβανóμενα; a certain preference for this word, as a label for preliminary lemmata that are presented before the main body of a treatise, is attested for Archimedes, though he is not ~ completely consistent in his usage of the word. In Prop. 17, the word λἆμμα will be used.
200.
III, 12.
201.
Note the change of tense. This could be an indication that the statement about KL intersecting AC was originally not part of the ekthesis, but of the proposition. As indicated by the way I set up the paragraphs above, I think that the whole text from “and the straight line” to “AB (is) to BC” is the proposition. Prop. 13’s claim thus has two parts: (i) AKDC is a trapezoid, i.e., KL and AC meet (in E), (ii) AE:EC = AB:BC. See the commentary for a defense of my decision. It has con sequences for the converse of Prop. 13 as well. For the converse can then assume both AK ǁ CD and AE:EC = AB:BC (even if the former condition is not explicitly mentioned), and derive K—L—D—E from there. The converse will be used in Props. 15 and 17.
202.
That AH ǁ CD and that therefore AKDC is a trapezoid will be shown in the first part of the apodeixis.
203.
This means that E is a point of similarity.
204.
Hultsch brackets the phrase “when CD is joined” as an interpolation (210, 8 app. Hu). For on his reading, the line CD is mentioned already in the ekthesis, and should not occur in the proof. See the above footnote and the commentary for the reconstruction of the overall argument in Prop. 13.
205.
Here Hultsch (perhaps unnecessarily) brackets the words “at L” (210, 10 + app. Hu), and in the following line “ἒχoυτα”(210, 11 + app. Hu).
206.
VI, 7.
207.
I, 27. Now we have shown that CD is parallel to AH. Therefore, we indeed have a trapezoid AKDC, with AH > CD; therefore, AC and KD (=KL) meet, and we call the point of intersection E. See the commentary.
208.
VI, 4, V, 16.
209.
Hultsch 211, # 1, claims that AH ǁ CD can be shown exactly as above, from ΔLHK ~ ΔDCL. However, that similarity rested on the assumption that D, K, and L lie on a straight line, and this is exactly what the converse is about to prove. In my opinion, we rather have to assume that DC is a parallel to AH in the converse. See the commentary.
210.
AK = AB and CD = BC, as radii of the respective circles.
211.
By assumption.
212.
With D — T — C.
213.
AE:AK = EC:CT (VI, 4); apply V, 16.
214.
N — D — C.
215.
VI, 4 and V, 16, as above.
216.
Again, it seems apparent that one must assume that CD ǁ AK (compare the above footnote).
217.
I, 34.
218.
CN:CD = AE:EC by assumption. CN = CD + DN, and AE = EC + AC; therefore, DN:CD = AC:EC (V, 17).
219.
V, 16.
220.
CDN is a straight line ⇒∠CDK + ZKDN = π. It was shown above that ∠EDC = ∠NDK; therefore, ∠CDK + ∠EDC = π.
221.
AE:EC = (AB + BE):(EZ + ZC) = AB:CZ (assumption in the converse) ⇒ BE:EZ = AE:EC (V, 19); AC:EC = DK:DE (VI, 2), and thus AE:EC = KE:ED (VI, 1).
222.
VI, 1, height EL.
223.
VI, 1, height BE.
224.
III, 36.
225.
The diameters BC and BD of the semicircles are assumed to be in line. E and H are the points of touch with the third circle. There are three possibilities for the relative position of the semicircles and the circle involved (configurations 1–3, cf. figures a–c). Even though only two of these configu rations are needed for the arbelos theorem, the author of the little treatise gives a complete account of the lemmata involved. See the commentary on the Archimedean features of Props. 13–18.
226.
The Greek word is “ὑπερoχή” (excess); as in Prop. 9, it is translated as “difference”.
227.
The technical Prop. 14 yields the central result needed for establishing the arbelos theorem. Specifically, it is the intermediate step labeled as “*” below that is most important for the follow ing theorems. See the commentary on Archimedean features of Props. 13–18, and compare the footnote on “*.”
228.
Hultsch (p. 215, # 1 Hu) provides a proof involving an auxiliary construction, and reference to Prop. 13. Instead, one could simply assume implicit appeal to an elementary step of inference, capturing the same content as the group of theorems in Coll. VII, 102–106 mentioned in the foot notes to Prop. 8: Whenever one has a configuration with parallel chords in tangent circles, the lines connecting the endpoints “crosswise” also go through the point of touch. Another possibility for this step in Prop. 14, though valid only for configurations 1 and 2, would have been to appeal to a theorem as in Lib. ass. I. See also Co p. 74, A.
229.
Again, we may have an appeal to a theorem about parallel chords in tangent circles (cf., e.g., Coll. VII, 102–106; compare the preceding footnote).
230.
Both triangles have a right angle, and they have the angle at B in common.
231.
I.e., the area of the rectangle with sides CB and BK. The Greek text has “τò ὑπò ГB BK περƖεχóμευoυ χωρίoυ.” The fact that we are dealing with areas is explicitly emphasized, and this seems to be a peculiarity of the text in Props. 13–18. It may very well go back to the style of the original author of the treatise.
232.
The Greek phrasing is “τò ὑ πò HB BΘ.” This is also different from the abbreviations “τò ὑπò HBΘ” and “τò ὑπò τῶυ HBΘ”, which are taken, in this translation, as technical formulae for the rectangle HBT. The expression “τò ὑπò HB BΘ” is elliptic for “τò ὑπò HB BΘ περƖεχóμευoυ χωρίoυ.” The translation will keep track of this differentiation by adding the phrase “area comprised” in brackets throughout Props. 13–18. Compare the preceding footnote. BT:BK = BC:BH (VI, 4) ⇒ BT × BH = BC × BK (VI,16).
233.
The argument is completely analogous to the one in the preceding step. The triangles are similar because of the right angles and the common angle at B; similarity implies the stated proportion (VI, 4), and thus (VI, 16) the equality of the rectangles.
234.
III, 36 for configurations 1 and 3; III, 35 for configuration 2.
235.
We have shown: CB × BK = HB × BT, DB × BL = ZB × BE, and HB × BT = ZB × BE. Therefore, CB × BK = DB × BL.
236.
In that case, DB = BL. The limit case will be used in Prop. 17 and may have been inserted here precisely for that purpose (cf. 214, 20–216, 1 app. Hu).The passage framed by “*” is the core of Prop. 14. Its result will be quoted several times in what follows, independently of Prop. 14 itself.
237.
VI, 16.
238.
(BC + BD):BD = (BL + BK):BK (V, 18); (BC - BD):BD = (BL − BK):BK (V, 17); therefore: (BC + BD):(BC − BD) = (BL + BK):(BL − BK) (V, 22).
239.
In numbers, BM is the arithmetic mean of BK and BL; the author of the arbelos treatise avoids using terms coined for numbers to label properties of magnitudes.
240.
The equality of the rectangles mentioned was shown in * above.
241.
VI, 16.
242.
(LB − BK):BK = (CB − BD):BD (V,17); from KL:KB = CD:DB we therefore get KL:(LB − BK) = CD:(CB − BD) (V, 22).
243.
LB = LM + MB; BK = MK − MB = LM − MB. From here, it follows immediately that BM:radius = CD:(CB − CD). Instead of giving the result for configurations 2 and 3 explicitly, and then restating it in the summary of what has been shown, the text proceeds directly to the summary.
244.
The triangles are both similar to the triangle BHC.
245.
VI, 4 and VI, 16.
246.
By construction: TK = ZL = AM, because TZ is a diameter parallel to BC.
247.
From *, we get: BC × BK = BD × BL ⇒ BC:BD = BL:BK (VI, 16); first configuration: BD + CD = BC, and BK + KL = BL; thus: CD:BD = KL:BK (V, 17); second and third configuration: BC + BD = DC, and BL + BK = KL; thus: CD:BD = KL:BK (V, 18). From these equations, it follows in all three cases that BC:CD = BL:KL ( V, 16 and V, 22).
248.
VI, 16.
249.
The argument is analogous to the preceding one. From *, we get: BC × BK = BD × BL ⇒ BC:BD = BL:BK (VI, 16); first configuration: BC = BD + CD, and BL = BK + KL; thus: CD:BD = KL:BK (V, 17) second and third configuration: DC = BC + BD, and KL = BL + BK, and DC:BD = KL:BK (V, 18), also. In both cases, V, 16 yields BD:CD = BK:KL.
250.
R, T’, and T are the points of touch with the semicircles over BD and BC and the first added circle EHT respectively. M and N lie on BC.
251.
Again we get three possible configurations, on the basis of the configurations in Prop. 14. Each of them leads to exactly one possibility for the second circle to be inscribed into the respective configuration. Note that in Hultsch’s edition, configuration 1 from Prop. 14 leads to configuration 1 in Prop. 15, whereas configuration 2 leads to configuration 3, and configuration 3 to configuration 2. I have numbered the figures in concurrence with Prop. 14. In A, the second diagram for Prop. 15 concerns a limit case that is not treated in the text, but relevant for Prop. 17 and Addition 2 to Prop. 16. For a correct diagram and a reconstruction of the proof for the limit case see appendix Hu p. 1227 f.; cf. also Co p. 78 P. The figure for the limit case given in A is reproduced in an appendix to this edition. The figure for the configuration that results from building on configuration 3 has been added here; it is modeled on Hu, since it is missing in A. The occurrence of the figure for the limit case indicates a loss of text that was originally part of the source at some stage in the transmission.
252.
Z is taken as the point of intersection between BZ and AP.
253.
I.e.: CB + BD.
254.
Prop. 14.
255.
I have taken “καὶ ἡ BM πρòϛ…” (220, 1) and “καὶ ἡ BN πρòϛ…” (220, 6) to be syntactically parallel. We get two corresponding statements about line segments cut off by perpendiculars in relation to radii of corresponding circles. Prop. 14 can be applied directly for BM:AT and NB:TP.
256.
MB:AT = NB:TP, because they are both equal to either (CB + BD):CD (configuration 1) or to (CB − BD):CD (configuration 2/3). V, 16 yields MB:BN = AT:TP. This proportion will be used again in the course of Prop. 15.
257.
X is the point of intersection between MZ and PN. For configurations 1 and 2, consider ΔZBM with intersecting line PX, parallel to BZ. We get: BN:NM = ZX:XM (VI, 2); this transforms to NM:BN = XM:ZX (V, 16), and thus: BM:BN = ZM:ZX (V, 18). For configuration 3, consider DMNX, with intersecting line ZB, parallel to NX. We get: BN:BM = ZX:MZ (VI, 2), and V, 16 yields BM:BN = MZ:ZX. PN ǁ AM by construction, and therefore: ZM:ZX = ZA:ZP (VI, 4, with V, 16, for ΔZMA, ΔZXP). This argument is applicable in all three possible configurations, and we get: BM:BN = AZ:ZP.
258.
Having shown BM:BN = AT:PT and BM:BN = AZ:ZP (cf. preceding footnotes), we get: AT:PT = AZ:ZP.
259.
We use the converse of Prop. 13 to establish that E, R, and Z lie on a line. Hultsch (222, 7/8 app.) believes the reference to Prop. 13 is due to an interpolator.
260.
Prop. 13, Addition.
261.
III, 36.
262.
In ΔOPT and ΔSAT, ∠A = ∠P, because PO ǁ AS (I, 29). Since the triangles are isosceles, they are similar. Therefore, they have equal angles at T. Since PA is a straight line, ∠STA +∠ATO = π, and A — T— O is a straight line.
263.
∠BZT = ∠OPT, because NZ ǁ PO (I, 29); above, it has been shown that BZ = ZT; obviously, PO = PT, too; therefore, ΔBZT ~ ΔOPT, and TO must pass through B. Otherwise, ΔBZT would not be isosceles.
264.
An argument analogous to the one showing that TO must pass through B was used in the converse to Prop. 13. Hultsch suspects the reference to Prop. 13 to be an interpolation (222, 7/8 app. Hu). It seems also possible that the arbelos treatise as a whole was taken out of a larger treatise, with a more substantial preliminary part, of which only Prop. 13 survives. See the commentary on Prop. 13.
265.
ΔBMK ~ ΔBNP by construction ⇒ BK:BM = BP:BN (VI, 4), and BK:BP = BM:BN (V, 16).
266.
We have shown above: AZ:ZP = AT:PT = BM:BN. Now we also have: BM:BN = BK:BP. Obviously, AT = AS and TP = PO (radii). Therefore, AS:PO = ATPT = BK:BP. Consider that ΔBKS ~ ΔBPO. BP:PO = BK:KS (VI, 4) ⇒ KS:PO = BK:BP (V, 16). It follows that AS:PO = KS:PO, and AS = KS must hold (V, 9).
267.
AK = AS + SK = 2 AS = 2KS. Consider the pairs of similar triangles BKM, BPN and BKS, BPO; We get: PN:KM = BKBP = KS:OP, and thus (V, 16): PN:OP = KM:KS. Therefore, PN:2OP = KM:2KS = KM:KA. 2OP is the diameter of the circle TRT’, and KA is MA + AK = MA + the diameter of the circle EHT.
268.
Some bit of text has been lost at the end of Prop. 15 (cf. 224, 11 app. Hu). As said above, the manuscript A has a figure for the limiting case of Prop. 15 (the case used in Prop. 17 and in Addition 2 to Prop. 16), but no argument. For such an argument, cf. Co p. 78, Lemma in P, and appendix Hu p. 1227 f.
269.
In the arbelos proper, each circle in the sequence touches the semicircles over BD and BC, and its own predecessor in the sequence. For the first circle, this is the semicircle over DC.
270.
Prop. 14, intermediate step *.
271.
VI, 16.
272.
Prop. 14, Addition 1.
273.
Prop. 15.
274.
It was shown in the first part of Prop. 16 that ZT = AM.
275.
(AM + ZT):ZT = 2 ZT:ZT = PN:diameter of circle P.
276.
PN:diameter ~ 2:1 ⇒ PN + diameter:diameter ~ 3:1.
277.
Prop. 15.
278.
The argument in Prop. 16 is related to complete induction. See the commentary.
279.
The first step of the induction is then trivial. The argument can proceed from there, on the basis of Prop. 15, as in Prop. 16. One has to assume the limit case of Prop. 15, for which only the figure, but not the actual argument survives (see notes above). Co p. 80 F provides a direct argument without reference to Prop. 15.
280.
The squares are to have a ratio like two square numbers.
281.
I.e.: Prop. 17 will show hat DZ²:d(circle A)² = BC:CD. Because DZ = AM, one can see (e.g., using X, 9) that AM will be commensurable in length with d(A) iff BC has to DC a ratio expressible in square numbers.
282.
Hultsch (230, 4–8 Hu + app.) believes the example is due to interpolation.
283.
BC = 4CD ⇒ DZ² = 4 diameter²; this entails DZ = 2 diameters.
284.
Prop. 15.
285.
Here, the author uses the word “λήμμα”; before, in the introduction to the arbelos treatise, the word “λαμβαυóμευα” was used.
286.
D is on BC. B — D — C.
287.
Appeal to theorems like Coll. VII, 102–106 (tangent circles, parallel chords and lines through the point of touch) seems most likely. ZAB and TAD will be straight lines, because TZ is parallel to BC, and A is the point of touch. As above in Prop. 14, Hultsch ad locum comments that this could be shown via Prop. 13, and refers to his footnote on Prop. 14 to this effect. Co p. 81 B refers to his Lemma p. 74 A.
288.
Prop. 14, passage *. As noted above in the footnotes to the passage, Prop. 17 uses a limiting case for the result in passage *, which was probably included there with a view to Prop. 17.
289.
BC:BD = BD:BK (VI, 17) ⇒ (BC − BD):(BD − BK) = BC:BD (V, 19), i.e.: CD:DK = BC:BD ⇒ BC:CD = BD:DK (V, 16). TZ = DK by construction, thus: BC:CD = BD:TZ.
290.
ΔBAD ~ ΔTAZ, because TZ ǁ BD and B — A — Z, D — A — T are straight lines (I, 29). Therefore, BD:DA = TZ:TA (VI, 4), and BD:TZ = AD:TA (V, 16).
291.
In the right-angled triangle TZD with height AZ, we have: TZ:TA = TD:TZ, and ZD:TD = AD:ZD (VI, 8); therefore: TZ × TZ = TD × TA, and ZD × ZD = TD × AD (VI, 16). Therefore: (TZ × TZ):(ZD × ZD) = (TD × TA):(TD × AD) = TA:AD (VI, 1). An explicit use of abstract duplicate ratios, interpreted as ratios of squares, as suggested by Hultsch here (p. 233 Hu), can be avoided. Compare also Co p. 81, F.
292.
We have shown above: BC:CD = BD:TZ = DA:TA, and finally: TA:AD = (TZ × TZ):(ZD × ZD). Apply V, 16.
293.
This statement may be an indication that Pappus himself thought that there were at least two layers present in the source he is using.
294.
A — E — C.
295.
Prop. 15.
296.
Prop. 15.
297.
Perpendicular from T: radius ~ 3:1; therefore, (perpendicular from T + 2 radii): 2 radii ~ 5:2. Note that in this phrasing, numbers and magnitudes are again kept apart conceptually.
298.
This is, again, an argument by (complete) induction. Here the odd numbers are viewed as an infinite sequence, in ratios.
299.
This statement is misleading. According to the proem of Archimedes’s Spiral Lines, it was Archimedes himself who proposed the theorem, challenging Konon to prove it. When the latter died before being able to seriously attempt the task, Archimedes proceeded to publish his own treatment, referencing Konon as the original addressee and intended discussion partner.
300.
The Greek term γέυεσƖϛ means coming-to-be, creation, growing. It is used in every generation of a motion curve in Coll. IV, and I have left it untranslated.
301.
Note that the circle is given from the start, and the spiral inscribed in it. In Archimedes’ Spiral Lines (SL), the spiral is created from two given motions, and the circle is described afterward around it. Only the version in Coll. IV will yield the angle section and the squaring of a circle. See the commentary on SL versus Coll. I V, and on symptoma-mathematics of motion curves.
302.
κεκήήσθω. Throughout the descriptions of the motion curves, Pappus will use either κƖυεῖυ or ϕέρεεθαƖ. Perhaps the two terms have a slightly different meaning. For lack of examples it is not possible to determine what the difference would amount to. I have chosen to render “κƖυεῖυ” with “move”, and ϕέρεεθαƖ with “travel”. In ordinary usage κƖυεῖυ is the broader term, whereas ϕέρεεθαƖ is restricted to locomotion.
303.
Rotation is in all likelihood clockwise, though counterclockwise rotation is possible, too. The synchronous linear motion is “inside out.”
304.
The two motions have to be synchronized, using the ratio of radius to circumference of a circle, i.e., π. The implicit inclusion of π is the reason why this version of the Archimedean spiral can be used to divide an angle in any given ratio (cf. Prop. 35), and also to square the circle (invoking SL 18). It also creates problems for the conceptualization of this version of the spiral. See the commentary.
305.
Reading A’s “αυτη” as “αὒτη”; both Hultsch and Treweek prefer “αὑτή” (234, 18 Hu; 101, 7 Tr).
306.
αρχƖκóυ σύμπτωμα. The word “archikon” implies the idea of “original” as well as “principal.” In fact, the main, property of the curve, the one on which the mathematical argumentation draws, stems directly from the curve’s origin. A similar use of αρχƖκóυ can be found at 252, 21 Hu for the quadratrix. The word symptoma originally denoted a chance happening or casualty. Within Hippocratic medicine, it was used to label the signs (symptoms) of a disease, the observable char acterizing property of a subject of study, the one the expert will look for and work with. Drawing on this scientific usage, it is then used in geometry for characterizing higher curves, and sometimes even conic sections. It obviously plays the role of a technical term; and I have left it untranslated. The symptoma of the spiral here, expressible in strictly mathematical terms, derives directly from the genesis, from the origin of the curve. The subsequent mathematical arguments, however, use the symptoma as a principle in the mathematical argumentation, as a quasi-definition of the curve, avoiding any reference to the genesis. For the significance of this move see the commentary on symptoma-mathematics.
307.
Compare SL14 (together with SL 2), for a spiral with circumscribed circle.
308.
Pappus’ explanation here is not very felicitous. It unduly fuses concepts of motion and speed. Nevertheless, Knorr (1978a, p. 50 f.) goes too far in concluding that Pappus misunderstood the whole mathematical context.
309.
Line segments between the center of the original circle and the spiral will be called “spiral radii” here.
310.
Spiral radii corresponding to equal angle increments form an arithmetical sequence. Compare SL12 (with SL 1). The property follows directly from the genesis. For an elementary argument that the spiral radii for equal increments of angles form an arithmetical series see Co p. 83 commentary on Prop. 20.
311.
Areas contained by the spiral line and the spiral radius at some point of the rotation will be called “spiral areas” here. Prop. 21 addresses the spiral area for the first complete rotation.
312.
Compare SL 24 (for a spiral with circumscribed circle); note the difference in argumentation. Prop. 21 uses quasi-indivisibles, whereas SL 24 has a classical proof via double reductio, and uses a progression of spiral radii (SL 12 (with SL 10), cf. Prop. 20). See the commentary.
313.
The ratio for the division is not specified. Most likely, it is 1:2ⁿ.
314.
Adopting Tr’s emendation KL for A’s KA (Tr 101, 26).
315.
Symptoma: AB:BZ = AB:BH = circumference:arc AC. Thus, AB:(AB − BH) = circumference: (circumference-arc AC) (V, 19).
316.
The path of reasoning in this somewhat lengthy sentence is rather straightforward. Because (AB:AH =) BC:CZ = circumference:arc CA (due to the spiral), while we also have circumference:arc CA = PK:KR (by construction) = LK:KW (VI, 2 and V, 18) = T’R:RW (VI, 4; V, 16; V, 18), we can infer that T’R:RW = BC:CZ.
317.
BC:CZ = T’R:RW implies BC:BZ = T’R:T’W (V, 19, addition). Then the stated proportion holds for the squares (VI, 22).
318.
XII, 2 (circles have the ratio of the squares over the diameters). The sectors in Prop. 21 are the same parts of their respective full circles (use VI, 33 and V, 15).
319.
XII, 11 (cylinders of equal height have the ratio of the circles at their base), and XII, 2 (circles have the ratio of the squares over their diameters).
320.
The manuscript A has a plural (σχήματα, 238, 17 app. Hu). Perhaps we do have a scribal error here, but it is also possible that Archimedes viewed the spiral area in this argument as actually composed of spiral sectors with quasi-indivisible arcs.
321.
Archimedes uses an argument that could be called “exhaustion” in the literal sense. It closely resembles arguments via indivisibles. See the commentary.
322.
XII, 10; Knorr (1978a) p. 55) notes that the reference to XII, 10 leaves a gap. II does not cover the implicit convergence argument for the spiral-figures, which is, however, crucial here.
323.
The addition targets a spiral segment with circumscribed circle. Thus, it is the true parallel to SL 24. The labeling of the spiral is “outside —in”, in contrast to the description in Prop. 19.
324.
The Greek term for the argument in Prop. 21 is indeed apodeixis (cf. 238, 26 Hu), suggesting that Pappus may very well have considered it as more than just heuristic exploration.
325.
ἱστoρίαϛ ἅϛƖoυ; the word ἱστoρία does mean “history,” among other things. Its original meaning is “investigation”, or “research.”
326.
Prop. 22 does not specify whether the circle is circumscribed or the spiral inscribed in a given circle. In both cases, we have a contribution to the symptoma-mathematics of the spiral. In the former case, the theorem would in addition be on a par, conceptually, with the theorems in SL. The fact that the circle is mentioned but not used in the theorem may be an indication that we still deal with the inscribed spiral. As in Prop. 22, and in difference from the description in Prop. 19, the spiral is labeled “outside-in.”
327.
XII, 2.
328.
Theon’s addition to VI, 33 (circles to sectors as circumference to arcs).
329.
Prop. 19.
330.
Prop. 22 uses Prop. 21 and the symptoma to express the desired ratio of spiral areas as a composite ratio. It is the composite of a ratio of squares over radii and one of radii, and this is declared to be equivalent to a ratio of cubes. The interpretation of composite ratios as quasi-products was not without its difficulties, though Archimedes seems to have used composite ratios that way without qualms cf. Saito (1986). Co p. 85, A refers to an Archimedean theorem on centers of gravity for solids.
331.
Spiral area BA:spiral area BZ = (circle BA:circle BZ) × (BA:BZ) = (BA²:BZ²) × (BA:BZ) = BA³:BZ³.
332.
Prop. 19 yields AB:BZ:BN:BE = 4:3:2:1. From Prop. 22, we see that corresponding full spiral areas are as 64:27:8:1. Subtracting the preceding spiral sectors at each stage, we get 37:19:7:1 for the spiral quadrants.
333.
For information on Nicomedes see the commentary. He is associated with the conchoid and the quadratrix (i.e., two of the prominent motion curves used for symptoma-mathematics in Coll. IV).
334.
δoθέυ this is the same term used in geometrical analysis. See the commentary.
335.
The distance CD is kept equal throughout the “dragging process” (neusis-property of the curve); DE (corresponding to the pulling rope for a ship) is variable.
336.
Accepting Hu’s addition of πεσεῖταƖ (244, 11 + app. Hu), although Tr may be right in preserving the manuscript reading (Tr 105, 11).
337.
The symptoma seems to be read off a curve already drawn, not abstracted from the generating motion (as was the case for the spiral). See the commentary.
338.
The spelling of the Greek name for the curve appears as κoχλoεƖδήϛ in A and in Hu’s text through the end of Prop. 25. In almost all occurrences in A, however, the λ was expunged later, and a λ super scripted, changing the name to κoγχoεƖδήϛ. It will be rendered as “conchoid” here.
339.
No documents about Nicomedes’ theorems on the other conchoids survive.
340.
Greek for “with an instrument”: “óργανƖκῶϛ”. This term should be differentiated from the stan dard Greek term for “mechanical”: μηχανƖκῶϛ. Co translates “instrumentaliter” (cf. at Co p. 89). For the significance of this difference see the commentary. What Pappus gives here is not a com ment on the conchoid itself, as “mechanical,” i.e., generated by motions, but a reference to the use of a concrete instrument, a “conchoid-compass”, to draw the curve. Such a compass can be easily constructed from the description of the generation of the curve via motions (cf. Eut., Comm. in Sph. et. Cyl. II, pp. 98, 1–100, 14 Heiberg).
341.
We do not have a treatise by Pappus with this title. Information on Diodorus and a work on the Analemma is also very scarce (cf. Heath Vol. II, p. 286 f.). Hultsch p. 246 ad locum suspects a corruption of the text, and offers “lemma 1” or “lemma 21” as possible readings.
342.
This side remark documents that Pappus must have been aware of the connection between the angle trisection, the duplication of the cube, the neusis construction for which the conchoid operates like a compass, and typical solid problems in general. See the commentary, and Props. 31–33, 42–44.
343.
For a discussion of neuses and their role in Greek mathematics see the commentary. In Coll. IV, neuses are also put to use in Props. 31–34, and in Props. 42–44 (picking up a reference in the meta-theoretical passage between Prop. 30 and Prop. 31).
344.
The Greek text has δoθείση, the term used in geometrical analysis. This suggests an analytic- synthetic background for the neusis and the conchoid (as a locus curve). Prop. 23 corresponds to Eut. In Arch. Sph. et Cyl. II, pp. 102–104 Heiberg. Compare also the apparatus in Hu ad locum, for parallels and doublets in Coll. III, pp. 58–60.
345.
D on CT, TD = given line.
346.
Probably Co p. 87 is right in suggesting “(straight) line AH” for “line EDH.” Then the procedure by trial and error makes sense, and one avoids having to draw out the conchoid. For once the conchoid is drawn, trial and error is no longer needed, and the sense of Pappus’ remark becomes unclear. The use of the term given may suggest an analytical context for Nicomedes’ original considerations. Co p. 87 nevertheless justifies the success of the ruler manipulation construction with the conchoid.
347.
The Greek text has “μóυηυ,” Hu 246, 22 emends to “μóυoυ,” following Co, and Tr emends as well. The only conceivable sense one might make of the manuscript reading is for Pappus to indi cate that Nicomedes furnished a single neusis construction, covering both the angle trisection and the cube duplication, whereas Pappus quotes the apodeixis of it, adapted to the case of two mean proportionals. Then Prop. 24 would still be essentially by Nicomedes, and Pappus would not claim more than his adaptation of it for the cube duplication here. This would diminish an apparent inconsistency entailed by the emended text: that Eutocius reports much the same neusis construc tion as Nicomedean, whereas in the emended text version Pappus seems to claim it for himself. See also the following footnote, and the commentary. Perhaps the manuscript reading could have been defended, then. Since this is a question of a single letter only, though, I follow the authority of the editors. In any case, the mathematical sense is not affected, and the majority of scholars ascribe the content of Prop. 24 to Nicomedes, even in face of the phrase in the emended text.
348.
Eutocius reports the very same argument in In Arch. de Sph. et Cyl. 104–106. Perhaps Eutocius is quoting from Pappus; cf. Ver Eecke (1933b, p. 188, # 3). Jones (1986a) considers the possibility that Eutocius draws on a report by Pappus in Coll. VII.
349.
II, 6.
350.
I, 47.
351.
I, 47.
352.
VI, 2 with V, 16 (ΔMBL ~ ΔMAL, ΔMBK ~ ΔLCK, on parallel lines).
353.
ΔADL ≅ ΔBDH (I, 26), therefore HB = AL (=BC).
354.
V, 15: MA:AB = BC:CK implies MA:1/2AB = 2BC:CK.
355.
VI, 2 (ΔHZK ~ ΔCTK on parallel lines).
356.
V, 18.
357.
By construction (neusis).
358.
V, 9.
359.
II, 6.
360.
VI, 16.
361.
VI, 4 (ΔMBK ~ ΔLCK).
362.
VI, 4 (ΔMAL ~ ΔMBK).
363.
Once again, note the occurrence of derivatives of the technical term δoθέυ (250, 26, 27, and 28 Hu).
364.
a and c stand in the triple ratio of a:b (V, def. 11); cube numbers have two mean proportionals, and
365.
The Latin word “quadratrix” (i.e., squaring line) translates the Greek name (τετραγωυƖϛoυσα) for the transcendent curve that will be the subject of Props. 26–29. The Latin version is commonly used as the standard name for this particular curve, though the term can have other meanings, too.
366.
συνακoλoυθείτω; the basic verb is, once again “aακoλoυθέω” = follow along in order. As in the other instances in Coll. IV, it does not have the connotation of strict logical derivation — on the contrary (see below). On the use of “aακoλoυθεῖν” compare the remarks on analysis-synthesis in the introduction to Props. 4–12.
367.
This generation via synchronized motions is reminiscent of the genesis of the spiral in Prop. 19; the
368.
The quadratrix can be used also for the division of an angle in any given ratio (probably its origi nal use), and for problems related to this construction. Cf. Props. 35–38.
369.
The passage taken from Sporus differs significantly from the mathematical expositions in Coll. IV. Note, e.g., the rhetorical questions and the polemical style. Co p. 88 replaces the name “Sporos” with the Latin word “spero.” His paraphrase means: “I expect, however, that this line justifiedly and deservedly does not satisfy, for the following reasons.” The replacement changes the meaning of the introductory sentence, and indeed of the whole passage criticizing the quadra- trix considerably.
370.
The Greek text uses the third person singular. It is unclear whom Sporus’ argument targeted.
371.
The use of the notion “velocity” is not quite precise here. However, it is clear what Sporos means, and his argument is valid. In order to synchronize the two motions as required, one must know π — or else use an approximation to stand in for it. However, π is exactly what the curve is supposed to exhibit in construction. Co p. 88 paraphrases “motuum velocitates.” Hu 254, 7 emends A’s ellipti cal “aαναγκαῖoν.” For a parallel construction, without emendation, see, however 270, 11/12 Hu.
372.
The reading πῶϛoῖoνταƖ (how do they think) as given in A, was kept. Both Hultsch and Treweek reject it in favor of the reading πῶϛ oἶóν τε (254, 8 Hu + app/ Tr. 109, 11), attested in the minor manuscripts. Co p. 88 paraphrases “quo pacto arbitrantur.”
373.
Restoring A’s reading πρòϛ (when) instead of Hultsch’s πρò (= before; cf. 254, 16 Hu app).
374.
Restoring, with Tr 109, 20, the reading of A.
375.
For an extension of the quadratrix to the base line one needs to know the direction. As the quadra- trix does not have a constant direction, or even curvature, one needs, in the end, to know the position of H, and it would have to be determined beforehand, using the ratio of radius and cir cumference (π). My translation differs from Hultsch’s Latin interpretation. Co has the following Latin paraphrase, rejected by Hultsch (p. 89 Co): Sed ut cumque sumatur punctum …, praecedere debet proportio circumferentiae ad rectam lineam.
376.
The Greek word (δoθἆναƖ) is the technical term from geometrical analysis. It is not certain (in fact perhaps unlikely) that Sporus, whom Pappus paraphrases here, intended it that way. What is certain, however, is that Pappus is going to interpret it in this strict technical sense for Props. 28 and 29. See below, and see the commentary on Props. 26–29.
377.
Accepting Hultsch’s emendation où for the difficult manuscript reading ή, kept in Tr 109, 26. Co p. 89 keeps the manuscript reading, and paraphrases as a question: Or should we… ? The disad vantage is that in that case one would have expected the question particle at the beginning of the sentence.
378.
I.e.: accept it as fully geometrical. The quadratrix itself (in the motion description) is not fully accepted; but note the upcoming remark on the mathematics about it. It is quite possible that Sporus and Pappus have different opinions on this matter. The issue cannot be pursued here.
379.
Greek: μηχανƖκῴτεραν. This word, used for the curve itself here, and not just for the way in which it is generated, is different from the label “óργανƖκῶϛ”, i.e., “describable with an instru ment”. The latter was used in connection with Nicomedes’ conchoid (cf. footnotes above). Hultsch deletes the phrase “and it is put to use by the students of mechanics for many problems” as an interpolation (254, 24-256, 1+app. Hu). There is indeed no evidence that the quadratrix played a major role in mathematical treatises on mechanics. A similar phrase occurs at 244, 20 Hu. See the introduction to Props. 19–30 in the commentary on “mechanical.”
380.
Hultsch has changed the transmitted text considerably. His Latin paraphrase means: “but before I must report (assuming παραδoτἒoν) the problem that is solved on account of it.” With Tr 110, 1–2, I keep the transmitted text. Co’s paraphrase on p. 89 is compatible with this reading. See the commentary.
381.
Note the change of lettering in the diagram. Perhaps Prop. 26 was taken from a different source (Nicomedes, as opposed to Dinostratus, or else Sporus, for the curve’s genesis?).
382.
Note that the quadratrix is posited at the outset. The upcoming argument will keep the problem atical genesis of the curve out of sight, and use its symptoma only.
383.
This proportion will yield the construction of a straight line equal to arc DEB (Prop. 27).
384.
We get a classical proof via double reductio (so-called method of exhaustion). Apart from the (short and straightforward) alternative argument for the inverse of Prop. 13, this is the first, and the only, example for this argumentative technique in Coll. IV. On Prop. 26 see also Heath (1921, I, pp. 226–229).
385.
By assumption.
386.
This theorem is also used in Props. 36, 39, and 40, and a similar one in Prop. 30 (cf. notes ad locum). An explicit proof is given in Coll. V, 11 and Coll. VIII, 22. A possible justification might proceed as follows. XII, 2: circles have the ratio of the squares over their diameters; Circ. mens. I: circles have the ratio of the rectangles with radius and circumference as sides; V, 16 and VI, 1: circumferences have the ratio of diameters. V, 15: similar arcs have the ratio of diameters. The frequent occurrence of this motif may indicate that it is part of the special “jargon,” a kind of basic tool within the “analytic track” of symptoma-mathematics of the third kind. Specifically, it might be a typical tool of Nicomedes. Nicomedes apparently systematically exploited properties of spiral lines, taking Archimedean argu ments as a starting-point. Compare Pappus’ remarks on the study of spiral lines and quadratrices as a central branch of geometry of the linear kind in the upcoming meta-theoretical passage.
387.
BC:CK = CD:CK = arc BED:BC (assumption); CD:CK = arc BED:arc ZHK ⇒ BC = arc ZHK ( V, 9).
388.
arc BED:arc ED = BC:HL (symptoma). arc BED:arc ED = arc ZHK:arc HK (equal parts).
389.
arc ZHK:arc HK = BC:HL; arc ZHK = BC ⇒ arc HK = HL (V, 9). This is not possible, because 2HL is a chord under two times arc HK.
390.
Just as in the first part of the “exhaustion,” one gets: CD:CK = arc BED:BC (assumption); arc BED:arc ZMK = CD:CK ⇒ arc ZMK = BC (V, 9). arc BED:arc ED (= arc ZMK:arc MK) = BC:HK (symptoma).
391.
HK must be larger than arc MK. I am not aware of an elementary geometrical argument in ancient geometry for this (correct) statement. Hultsch and Ver Eecke (1933b) ad locum refer to an argument that can be reconstructed from (Ps.-) Euclid, Catoptrics 8.
392.
Construct the third proportional s for TC and CB (VI, 11): TC:BC = BC:s; TC:BC = BC:arc BD (Prop. 26 with V, 16) ⇒ s = arc BD. Then 4 s is equal in length to the circumference of the circle.
393.
Circ mens. I. This rectangle can be transformed into a square via II, 14.
394.
Here Pappus picks up the discussion before Prop. 26, on the generation of the quadratrix via motions and the mathematical status of the quadratrix.
395.
αναλúεσθαƖ; since this is a technical term, clearly referring back to the technique of analysis (cf. Props. 4–12, and 31 ff.), Hultsch’s Latin paraphrase “problema solvitur” does not capture the meaning and is in fact misleading. What is “analyzed” here is not the problem of squaring the circle, but the genesis of the quadratrix. Co paraphrases “lineae ortus … resolvi potest (p. 90). Both Prop. 28 and Prop. 29 provide a resolutio in the sense that they show that the quadratrix is given, if an Apollonian helix or an Archimedean spiral is posited (i.e., taken as given). See the commentary.
396.
With EZ: arc DC given, E will be shown to lie on a line that is determined relative to a certain helix, which is assumed as given. This characterization is independent from the genesis of the line via motions, which has been disqualified as conceptually inconsistent. It is not constructive, how ever, but rather a characterization via implicit relations. Note that the analysis is quite general in the sense that the ratio which is taken as given is not assumed to be the ratio of arc and radius, as in the quadratrix. Co p. 90, B is, in my view, mistaken when he assumes that. For each given ratio, the analysis shows that a unique line is determined by it via the intersection of the surface related to a given cylindrical helix and a given plane. For the special case of a ratio equal to arc ABC:AB, this line will be the quadratrix. Compare the end of Prop. 28, and Hultsch, * on p. 259 and #2 on p. 261.
397.
Co p. 90 D assumes, mistakenly in my view, that the height of the quarter cylinder constructed has to be equal to AB, and that the defining ratio of linear upward motion to rotation is that of arc ADC:BC.
398.
For a definition of the helix cf. Heron, def. 8, 1 and 8, 2.
399.
TD is perpendicular to the plane of the circular quadrant; it is now considered as the height of the cylinder under discussion.
400.
We create a rectangle BDTL, with E on BD and I on HL, and EI parallel to DT.
401.
This ratio is implicit in the helix as the relation of rotation and upward motion in its genesis.
402.
Data 8. The sentence is truncated in A. Above, I have translated the text as emended by Hultsch (260, 8–10 + app. Hu, see also #3 on p. 261 Hu). Tr 111, 27–112, 3 prints an alternative recon struction, closer to the actual manuscript reading, and therefore perhaps preferable (see the appa ratus in the Greek text).
403.
Data, def. 15. When a line is given in position, the parallel to it through a given point is said to be given as a parallel in position (para thesei).
404.
Data 41 and 29.
405.
The manuscript is severely damaged by water in this place, and the text is not legible (cf. the apparatus in the Greek text). Hultsch’s emendation “έν τέμνoντƖ ἅρα” leaves open the possibility that the intersecting plane is determined by EZ and ZI or else by BC and ZI, whereas Treweek’s emendation identifies the plane in question as the one determined by EZ and ZI. The version with BC/ZI as intersecting plane has the drawback that the endpoint Z of the intersection line is not uniquely determined. The plane has BC in common with the garland — shaped surface created by the helix. The version with EZ /ZI has the drawback that one would have to know the exact position of either EZ or ZI, and it is unclear how that could be accomplished at this stage of the analysis. I therefore prefer the former version. See the commentary.
406.
Severe damage to the manuscript text here; cf. apparatus to the Greek text for different emendations suggested.
407.
The point I lies on the line of intersection between the abovementioned plane and the surface created by the ascending line BC in the cylinder.
408.
E lies on the projection of the line created on the cylindrical surface onto the plane.
409.
ανελúθη. Compare the introductory phrase of Prop. 28 with note.
410.
The Greek text, again, has ἀναλúεσθαƖ. Compare the introductory phrase of Prop. 28.
411.
We are starting from a configuration with a section of a circle ABC and a part of it DBC. The arc ADC is not necessarily the arc of a quadrant (Co p. 91 is probably mistaken in assuming so). An Archimedean spiral will be assumed in it, and the analysis will show that any such configuration with spiral will determine a unique quadratrix-type line, though not necessarily the quadratrix itself. When a spiral is chosen with an inbuilt ratio equal to the ratio of the circumference of a quadrant to the radius, we get the quadratrix.
412.
Compare the description of the genesis of the spiral before Prop. 19. The direction of the travel through AB and through the circumference is reversed in comparison to the former version. Also, the spiral is inscribed not in a full circle, but in a sector. The above translation accepts Hultsch’s emendations in 262, 7–9. Tr 112, 17/18 prints Hultsch’ s version, but notes that one might have emended ΓΔA in 262, 7 Hu and kept the manuscript reading for the rest of the sentence. Then the spiral is generated exactly like the one in Prop. 19. In the Greek text, Tr’s suggestion was implemented (cf. apparatus).
413.
Symptoma of the spiral, following directly from the genesis.
414.
AB:arc ADC = BH:arc DC.
415.
By assumption.
416.
V, 9 or V, 15.
417.
This surface is built up over the spiral as limiting line of the base. Co p. 91/92 assumes a different situation, with a full cylinder quadrant and an inscribed Apollonian helix, in addition to the cylin-droid. For yet another reconstruction cf. Knorr (1986, p. 166 f).
418.
By construction, HK = BH, and ∠BHK = π/2. Therefore, ∠HBK = π/4.
419.
K lies on the line created by the intersection of the two surfaces mentioned, cylindroid over the spiral, and surface of the cone with vertex B.
420.
Without loss of generality, L and I can be chosen as the points of intersection between the parallel to BD through T and the straight lines EI, BL.
421.
The Greek word πλεκτoεƖδἆϛ (πληατoεƖδἆϛ in A, Tr 112, 27, and Ver Eecke ad locum) is used here as a technical term the context for which is now lost. Following Hultsch, I have left it untrans lated. What a plectoid surface looks like can be derived from the description given here by Pappus. There is no other, independent source.
422.
I lies on the line created by the intersection of the surfaces mentioned.
423.
Severe damage to the manuscript text; see the apparatus for different conjectures.
424.
The Greek text has ακoλoυθóν; once again, we have a context in which the word cannot signify a logical derivation, and must mean a next step in a somewhat orderly fashion. See the commentary on analysis-synthesis in the introduction to Props. 4–12.
425.
Although this introductory paragraph draws an explicit connection to Props. 19, 28, and 29, the path of reasoning about the spiral line is very different from Props. 28 and 29. It shows affinities to Prop. 21 (“meta-mechanical” path of reasoning about the motion curves, quasi-infinitesimals, limit process, no analysis).
426.
Compare the genesis of the plane spiral in Prop. 19. The ratio of the velocities for the two synchronized motions involved in Prop. 30 is simply 4:1. Cf. equations in polar coordinates: spherical spiral r = 1/4 ω, plane spiral in Prop. 19 ρ = (1/2π)ρ, plane spiral in SL ρ = aω, where a is a natural number or a ratio of two numbers. The spherical spiral by motions can be constructed in thought exactly.
427.
Cf. the full circle going through LOTI, intersecting the spiral in O. Co p. 93, C is mistaken in assuming that arc KL is fixed as a quarter circle now. A division 1:2n is likely (cf. Prop. 21).
428.
The symptoma of the spherical spiral is read off directly from the genesis via motions; cf. the plane spiral (Prop. 19) and the quadratrix (before Prop. 26), but contrast the conchoid (before Prop. 23). I have based the translation on Hultsch’s emendations in 264, 16/17 Hu. Tr 113, 20–22 prints an emendation that is closer to the manuscript reading and is perhaps preferable (cf. apparatus).
429.
The formulation of the protasis is analogous to Prop. 21. An area theorem is expressed in terms of numerical ratios. Cf. Prop. 16: a theorem on a sequence of ratios of lines is expressed in numerical ratios.
430.
∠ADC = ∠ZCD = π/2 (III, 18); ∠ACZ = ∠ACD = π/4 (ΔADC isosceles). AC² = 2AD² (I, 47). 2(sector AZC):sector ACD = AC²:AD² = 2AD²:AD² (XII, 2) = 2:1.
431.
The configuration investigated has been transformed to a situation of analogy between surface with surface “inside” and sector with segment “inside”; cf. Prop. 21’s use of a parallel auxiliary configuration with rotation cylinders, and investigation via parallel processes of continuous inscription.
432.
arc ZE:arc ZA = ∠ZCE:∠ZCA (VI, 33); ∠CDA = 2∠ZCA; ∠CDB = 2∠ZCE (III, 32 and III, 20) ⇒ arc ZC:arc ZA = ∠ZCE:∠ZCA = ∠CDB:∠CDA = arc CB:arc CA (VI, 33).
433.
Symptoma of the spiral.
434.
V, 15 (surface LKTsurface OTN = surface hemisphere:full surface ONT).
435.
Surface hemisphere = 2 maximum circle (Sph. et Cyl. I, 33); circle with radius TL:maximum circle = TL²:(radius hemisphere)² (XII, 2) = 2:1 (I, 47); ⇒ surface of hemisphere = circle with radius TL; Surface of sphere through O, N with pole T = circle with radius TO (Sph. et Cyl. I, 42:); ⇒ Surface hemisphere:surface ONT = circle TL:circle TO = TL²:TO²; cf. Co p. 94, K for a slightly different path of reasoning.
436.
By construction, TL = AC = ZC, and BC = TO as chords under equal arcs (III, 29).
437.
XII, 2; V, 15 (circles have ratio of squares over radii); the same proportion holds for equal parts. Ver Eecke (1933b, p. 204, #4) refers to Sph. et Cyl. I, 42/43 here.
438.
An implicit limit process is used (cf. Prop. 21). The sought areas are analogously enclosed between all circumscribed and all inscribed composite circular areas/spherical sections. By choos ing the arcs involved in the construction ever smaller, the desired lines and areas are approximated.
439.
Sph. et Cyl. I, 33 (surface of the complete sphere = 4 area of maximum circle). Sph. et Cyl. I, 35: surface hemisphere = 8 quadrants of maximum circle. Thus, surface above spiral = 8 segments.
440.
We compare the remainders after subtraction. Since surface above spiral = 8 segments, we get that surface hemisphere – surface above spiral = surface below spiral = 8ΔACD. 8ΔACD = 8(1/2 AD²) = 4AD² = (2AD)².
441.
Essentially the same statement about the three kinds of geometrical problems is found in Coll. III. This passage is somewhat of a locus classicus on methodology. In fact, it is only found in Pappus in this degree of generality. See the commentary.
442.
λένη (270, 3 Hu); since Aristotle’s theory of scientific argumentation (Analytica Posteriora), the word had been a standard technical term in Greek theory of science. It has a classificatory connotation (kinds versus species), but it is also used to denote the subject matter of a scientific discipline as a closed field of essential connections. A possible translation for genos is “class”, but this obscures the connotation of the word with concepts of kinship, and the connections with an established discourse on scientific methodology.
443.
λúεσθαƖ (270, 6 Hu); unlike “ana-luein” (Props. 28 and 29), this word means “solve.”
444.
Note that the classification of the kinds is derived from the objects needed for a constructive solution, i.e., mathematical lines, not from tools of construction and performance (e.g., ruler and compass).
445.
κατασκευή (270, 11 Hu), the technical term for the construction in a classical apodeixis.
446.
ὑπoλείπεταƖ (270,13 Hu), a hapax legomenon in Coll. IV. Perhaps it was Pappus himself who lumped all the rest of mathematical problems into one “kind.” See the commentary.
447.
έπƖπεπληλμένωέ (270, 17 Hu), perhaps related to the term πληκυoεƖδἆ ϛ/πλεκτoεƖδἆ ϛ in Prop. 29.
448.
Cf. Props. 28 and 29. The space curves created in the intermediate steps there belong to this group.
449.
λραμμƖκαὶ έπƖστασεƖέ (270, 20/21 Hu); probably a book title. There is no information outside Coll. III/IV available on Demetrius. Ver Eecke (1933b, p. 207, # 3) dates Demetrius roughly in the first century BC, because Menelaus (see below) lived in the first century AD; cf. also Tannery (1912, Vol. II, pp. 1–47).
450.
έπƖπλoκή (270,21 Hu), perhaps related to the participle “έπƖπεπλεγμένoϛ” used above. Philo of Tyana is otherwise unknown; Ver Eecke, 1993b p. 207, #4 dates him to the second century BC.
451.
έκ έπƖπλoκϛ πλεκτoεƖδῶ (or: πληκτoεƖδῶν) τε κ αƖ στερεῶ παντoίων; the reference here is certainly to a another book, though probably not directly to a book title. The surfaces used in Prop. 29 probably are examples for such “twisted plectoids”.
452.
A considerable corpus of contributions to the geometry of such “higher” curves must have existed.
453.
Menelaus of Alexandria, an astronomer of the first/second century AD, was a predecessor of Ptolemy. His attested works include three books on spherics (preserved in Arabic), a work contain ing tables of chords in circles, a work on hydrostatics, a treatise on the settings of the signs of the zodiac, Elements of geometry, and a work on higher curves, with connection, inter alia, to the duplication of the cube, and to positions of the fixed stars. Hultsch refers to Chasles, Aperçu historique for a possible reconstruction of the line called “the paradox” (cf. 271, #4 Hu).
454.
Note the plurals. Pappus has described general quadratrices in Props. 28 and 29. Examples for spirals are mentioned in Props. 19 and 28–30. He has mentioned the existence of several con choids in Prop. 23. The cissoid was originally invented by Diocles in the third century BC and apparently generalized later; cf. Knorr (1986, pp. 246–263). In Pappus’ text, the other curves are indeed labeled as types of spirals, perhaps because all such “higher” curves involve a rotation along with a linear motion.
455.
Apollonius classified plane and solid neusis problems, and differentiated them into two classes according to the lines needed for their constructive solution. He may have attempted to develop a complete operational toolbox to solve problems that would fall under those types, determining limiting conditions and providing a scale of increasing complexity via analysis. There is no clear evidence, however, that the demand of “keeping within the kind” ever reached the status of a fundamental claim with universality for all geometry, and all geometers. See the commentary.
456.
Hultsch (273, #1 Hu) believes this must be I, 52; Zeuthen (1886, pp. 280–288), Tannery (1912, vol. I, pp. 302–311) and others show, however, that it could have been the problem of finding the normal to a parabola (Con. V, 62 in Toomer’s 1990 edition). Apollonius treats it analogously to the (solid) case of the hyperbola and the ellipse. In the case of the parabola, however, a plane construction would have sufficed, if one takes the parabola in question as given; cf. Zeuthen (1886, pp. 286–288).
457.
A has a genitive (location) here (and in the parallel phrase in Prop. 44). Hultsch emends to an accusative (direction). I have translated the transmitted text.
458.
Pappus’ objection here, and even more so his upcoming arguments about the neusis in question (cf. Props. 42–44 with notes and commentary) have often been misconstrued in secondary literature. The remarks refer to SL 18 (subtangent to a spiral of first rotation is equal to the circumference), which invokes neuses from SL 7/8. These neuses, in fact all neuses in SL 5–9, are indeed solid in Pappus’ sense (see the commentary on Props. 42–44 on how far he is able to show this). Pappus also claims that Archimedes could have done with a plane construction for the theorem in SL 18. Whether he means that Archimedes could have used a plane neusis or that Archimedes could have used some other plane argument, instead of the neusis, in SL. 18, is unclear. Co p. 95, C, refers to Witelo, Perspectiva I, 128 for a plane construction. Since Witelo may very well have had indirect access to the Collectio in the thirteenth century (cf. Unguru 1974), this may be significant, and certainly Witelo’s suggestion deserves scholarly attention.
459.
τῇ ϕúσεƖ στερεòν úπάρχoν; Pappus ascribes an essential, internal character to mathematical problems. This is in line with the Aristotelian meta-theoretical framework and vocabulary he has been using in this passage, as testified inter alia by his use of the term “kind.” See the commentary.
460.
The neusis can obviously be constructed with the conchoid (cf. Props. 23–25), when one chooses A as pole, CD as canon, and EZ as distance. Perhaps this was what Nicomedes did. Note the relation to the neusis that figures in Prop. 24. It seems quite plausible that Nicomedes indeed proposed essentially a single (μóνην) construction for both problems. Cf. above, introductory remarks on Prop. 24.
461.
The analysis was probably added by Pappus to an older argument for the angle trisection that constructed the neusis without using conics. He may have excerpted the analysis from a source. In A, the figure for the analysis differs from the one for the synthesis. The manuscript text also shows signs of confusion and incoherent partial corrections (cf. apparatus to the Greek text). Treweek 117a documents the differences in a list. The existence of these differences supports the thesis about the subsistence of an older layer of argument in Pappus’ text. They might be used for further investigations. An independent Arabic version, purely synthetic, exists. See the bibliographical references for Props. 31–34 in the commentary. Hultsch has adjusted the lettering of the diagram and of the items used in the argument for the analysis to the features of the synthe sis, thus making Prop. 31 conform to regular practice in analysis-synthesis (272 Hu + app; simi larly: Co p. 96/97). I have followed him. Treweek 117, 6–118, 2 opts for a more cautious and conservative emendation.
462.
Analysis-assumption. For the structure of analysis-synthesis in general see the introduction to the commentary on Props. 4–12.
463.
This part of the analysis is non-deductive.
464.
The resolutio begins here.
465.
Data, def. 6.
466.
Complete the rectangle ABZ and apply I, 43.
467.
συντεθήσεταƖ. The synthesis begins here.
468.
Prop. 33.
469.
Con. II, 12, paraphrased above in the footnotes to the last part of the analysis. BZ × ZH = ZH × HL = CD × DA = BC × CD.
470.
ΔBZA ~ ΔCZE, because AB ǁ CE; BC:CZ = AE:EZ (VI, 2); ZB:BC = ZA:AE (V, 16/18). ΔAED ~ ΔZEC, because AD ǁ CZ; ZE:EC = AE:ED (VI, 4); ZA:AE = CD:ED (V, 16/18). ⇒ CD:ZH = ZB:BC = CD:ED.
471.
V, 9.
472.
ZH ǁ ED by construction, and we have just seen that ZH = ED.
473.
Prop. 32 is the only example in Coll. IV with a diorismos fully carried through, in the sense that all possible cases for a problem are covered. But see the commentary on plane sub-cases for this generally solid problem.
474.
The position of E is yet to be determined.
475.
A on the semicircle over DE with center H (III, 31).
476.
By construction.
477.
I, 5.
478.
III, 20.
479.
I, 29 (parallels ZE and BC with transversal BE).
480.
I, 9.
481.
I, 1.
482.
I, 9.
483.
For an alternative, using a plane neusis, cf. Heraclius’ construction, which is contained in Coll. VII, and also reported in Descartes (1637, pp. 387–389) (188–193 Smith/Latham). It is noteworthy that Pappus did not opt for this route here. See the commentary.
484.
This phrase introduces the analysis. Co p. 97 translates “resolvemus.” This is more accurate than Hu’s “solvemus” (277 Hu). Hultsch comments (277, #1 Hu) that a shorter constructive proof would have been possible via Con II, 4, though Pappus’ resolutio (!) has its merits, too. Such a construction would have been purely synthetic. Pappus’ argument here contains analysis and synthesis and serves as exemplary for the methods of argument in “solid” problem solving. Prop. 33 shows strong indications for a close connection to Apollonius’ lost analytical-synthetical solution. It is also very close to Coll. VII, #204 Hu, by Pappus (commentary on an analytical argument in Apollonius’ original Konika, Book V). See the commentary.
485.
Analysis-assumption.
486.
This part of the analysis contains an extension of the configuration and is non-deductive.
487.
Data 28 (for DT) and 26 (for HD); this sentence marks the beginning of the resolutio.
488.
Data 25 (for T).
489.
Con. II, 3: AC² is equal to the figure on HD, and AC = 2AD = 2DC holds. The “figure on HD” is the rectangle constituted of the diameter HD and the latus rectum (parameter) k. Note, however, that Pappus is in all likelihood not referring to the now extant version of the Konika. Compare the footnote on the end of the analysis.
490.
CD = AD has just been shown. ΔBAC ~ ΔTAD on parallels CB, DT; BT:TA = CD:DA (VI, 2). Apply V, 9.
491.
Data 26.
492.
Data 2.
493.
Data 27.
494.
Data 26: AD is given in length and position, AC is given in position. Since AC = 2 AD, AC is given in length as well (Data 2).
495.
Data 52.
496.
Con. II, 3, cf. above.
497.
Data 26 and Data 2.
498.
Data 57.
499.
ἀναλúεταƖ. The extant Konika, a revision of Apollonius’ work on conics by Eutocius (sixth century AD), are purely synthetic and do not contain analyses for the constructions provided. Pappus consistently speaks of Apollonius’ treatise on conics as an analytic work in Coll. VII, and in Coll. IV he handles all problems that are solved by means of conics via analysis-synthesis. Probably the Apollonian work Pappus worked with was analytic-synthetic. For a synthetic solution of the construction problem mentioned here by Pappus c.f. Con. I, 54/55. Note that Pappus does not mention Apollonius by name. This could mean that in his time, Apollonius’ (analytical) Konika were the standard reference work, parallel to the Elements.
500.
συντεθήσεταƖ. This is the beginning of the synthesis.
501.
Elem. I, 45.
502.
Con. I, 54/55.
503.
Con. I, 32.
504.
ΔBAC ~ ΔTAD on parallels BC, TD. Since BT = TA, i.e., BA:TA = 2:1, CA:DA = 2:1 (VI, 2), and DC = DA.
505.
AC² = HD × κ by construction. We have just seen that ½AC = AD = DC.
506.
Con. II, 1/2.
507.
Prop. 34 gives the essential part of an analysis for the angle trisection via solid loci in two versions. No detailed constructive apodeixes are offered. Pappus is probably drawing on pre-Apollonian treatments of solid loci, perhaps by Aristaeus, and may have an interest in portraying the Apollonian solution, which he presented in detail in Prop. 33, as the classic one in terms of meth odology, which nevertheless did not render older contributions utterly superfluous. Prop. 34a is a simplified version of Prop. 34b, using the Apollonian technical apparatus, and may very well be by Pappus himself (cf. Jones 1986a, p. 584). It is the simplest of the three solutions in Coll. IV (cf. Heath 1921, I, pp. 241–242; Zeuthen 1886, pp. 210–212). 34b shows clear traces of an older treatise on solid loci (see below for Prop. 34b, and see the commentary).
508.
κεκλάσθω. This word has also been used in Props. 11/12. No construction for the “bending” is offered. Obviously, it is equivalent to the angle trisection. In Prop. 34, the task of trisecting an angle AMC is assumed to have been reduced to the task of trisecting the arc over a chord AC.
509.
Prop. 34a only considers the case where ∠BCA is acute. For the other two possible configurations, one can argue analogously, cf. Co p. 100 f. and appendix Hu p. 1230.
510.
Any point B that meets the condition about the base angles lies on this hyperbola.
511.
ΔEBD ≅ ΔCBD (I, 4) ⇒ ∠BEC = ∠BCA (= 2∠BAE by hypothesis). ∠BEC = ∠BAE + ∠ABE (I, 32) ⇒ ∠BAE = ∠ABE, and ΔABE is isosceles (I, 6).
512.
Choose H on AC, with HC:AC = 1:3 (VI, 9).
513.
Data 2, Data 27.
514.
CZ = 3CD and AC = 3CH; 3HD = 3(CH − CD) = 3CH − 3CD = AC − CZ = AZ.
515.
I, 47; ED = EZ by construction.
516.
II, 6: DH × AZ + EZ² = AE², i.e., DH × AZ = AE² − EZ²; AE = BE was shown above.
517.
AZ = 3DH was shown above; VI, 1.
518.
Consider the converse of Con. I, 21 (not established as a theorem in itself). Con. I, 21 states that for all points B on the hyperbola through H with latus transversum AH, parameter 3AH and ordi nate angle π/2, the above equality holds. In the analysis, we can therefore “conclude” from the equality that B lies on this hyperbola. Note, however, that this justification via Apollonius’ Konikamay be anachronistic in the sense that the alternatives 34a and 34b may very well draw on a pre- Apollonian treatment of the angle trisection via “solid loci” perhaps by Aristaeus. See the commentary.
519.
By construction of H, AH = 2HC.
520.
VI, 9.
521.
Con. I, 54/55.
522.
Retrace the steps of the above analysis. All points B on the hyperbola have the property that 2∠BAC = ∠BCA. See the commentary for a sketch of the apodeixis suggested here. Co p. 101 gives an extended apodeixis, considering all three possible cases for ∠BCA.
523.
For a reconstruction of an angle trisection on the basis of the considerations given here see the commentary. Hultsch p. 285 # 3 refers to a discussion of the synthesis by Commandino (cf. Co pp. 101–102, O). The construction can, of course, be used for a number of “solid” problems, and that may be the reason why Co integrates a longer exposition of a constructive proof. See also the comments on Props. 42–44.
524.
This version is closely related to a (lost) argument from Euclid’s Solid loci, ultimately resting on a prior argument by Aristaeus. For it is closely connected to Pappus’ commentary on such an argument in Coll. VII (#237 Hu, Jones (1986a, # 316–318, pp. 365–369, with 583 f); cf. Zeuthen (1886, p. 215) for the connection to Aristaeus). See the commentary, and cf. Heath (1921, I, pp. 243–244, II, pp. 119–121), Zeuthen (1886, pp. 212–215), and Knorr (1986, pp. 128–137 and 327). Knorr expands on Zeuthen’s arguments.
525.
λóγoϛ. Hultsch translates “proportio”. i.e., “ratio”, probably the ratio 3:1. “Logos” can, however, also mean “account”, “argument”. This translation seemed preferable.
526.
Analysis-assumption.
527.
VI, 33.
528.
∠ACB = 2∠BAC by assumption; thus, ∠DCA = ∠DAC, and ΔADC is isosceles (I, 6).
529.
DE is the height in the isosceles triangle ADC (I, 26).
530.
Data 7, Data 27.
531.
ΔABZ ~ ΔADE, on parallels DE and BZ; AD:DB = AE:EZ (VI, 2/ V, 16). In ΔACB, ∠ACB is bisected by DC, with D on AB; AC:BC = AD:DB (VI, 3).
532.
I, 47.
533.
In Coll. VII, Prop. 237, (cf. Jones 1986a, pp. 365–369, # 316/317), a hyperbola is established via analysis-synthesis, the points of which satisfy the conditions derived in the above analysis. It is the hyperbola with focus C, directrix ED, and eccentricity 2. In the analysis here we can “conclude”: B lies on this uniquely determined hyperbola. See the commentary. Co pp. 102–103, E provides an alternative argument.
534.
Bisect AC in E, draw ED as a perpendicular onto AC, and describe the hyperbola with directrix ED, focus C, and eccentricity 2, using Coll. VII, 237. The hyperbola intersects the given arc AC in B, in which the arc is divided in the ratio 2:1. For the proof, retrace the steps of the analysis. See the commentary for a list of the decisive steps. For an alternative synthesis for the trisection discussed in Prop. 34b, including a separate treatment of all three possible configurations, see also Co pp. 103–104, E (starting at “et compositio manifesta est”).
535.
Apparently, Pappus believes that if an analysis leads to conics, one has shown that the problem in question is (in general) solid. But see the discussion of analysis as a criterion for the determination of problem levels in the commentary on Props. 42–44. Pappus is correct in his assertion that the angle trisection is solid, and his analysis does show that it is not linear (analysis demarcates sharply “upward”).
536.
The first of the arguments in Prop. 35 (via the quadratrix) targets acute angles.
537.
Arc KT:arc LT = KB:AE; arc KT:arc MT = KB:DH (symptoma) ⇒ arc LT:arc MT = AE:DH ( V, 16 and V, 22); DH = ZE by construction.
538.
Apply V, 17 to arc LT:arc MT = AE:ZE; AZ:ZE equals the given ratio by construction.
539.
The labeling BZDC suggests motion of the generating point from B to C, as in the genesis in Prop. 19. Rotation could be clockwise or counterclockwise. The labeling CB for the generator suggests a motion from C to B, in deviance from the description in Prop. 19.
540.
Divide DB in E in the given ratio (VI, 9).
541.
BC:BD = circle:arc AC; BC:BZ = circle:arc HC (symptoma) ⇒ BD:BE = arc AC:arc HC (V, 22). Co p. 105 G refers to SL 14 instead.
542.
V, 17.
543.
Analysis-assumption. Prop. 36 gives only an analysis, reducing the problem to the division of an angle in a given ratio. Then Prop. 35 is invoked.
544.
In the smaller circle, the arc over the same angle as AEB (arc CT in the figure) will be smaller than the arc CTD, which was assumed to be equal to arc AHD.
545.
XII, 2 with Circ. mens 1 and VI, 1 / V, 15 (similar arcs are in the ratio of the radii (or the circum ferences) ). Cf. the proof protocol of Prop. 26, section *. The same argument about similar circular arcs and radii was also used in Prop. 26 and will be used in Prop. 39 and 40. A similar argument was used in Prop. 30.
546.
Prop. 35. Co p. 106/107, F provides a constructive proof. See also the commentary.
547.
The problem in Props. 37 and 38 constitutes a generalization of the inscription of a regular pen tagon in I V, 10/11 (of the Elements). In analogy to the Euclidean construction, a triangle with the required ratio of angles is sought first, and then the polygon is put together from isosceles triangles. See the commentary.
548.
Analysis-assumption.
549.
Extension of the configuration for the analysis, non-deductive.
550.
In the problem; we are now in the resolutio.
551.
III, 20.
552.
Data 9.
553.
VI, 33.
554.
C is given, because it can be constructed using Prop. 35. From “C is given” one might conclude that the triangle is given in kind via Data 30 and Data 40. This is how I would prefer to read the reso- lutio. For an alternative explanation see Hultsch p. 291, * and Co p. 107 E. The phrase “C is given” appeared suspicious to him, and he suggests “the straight line BC is given in position” in its place.
555.
Cf. Data, def. 3 for given in kind. A triangle is given in kind when its angles are given.
556.
συντεθήσεσαƖ.
557.
Prop. 35; the proposition is directly applicable only for angles that are at most right angles. Otherwise, divide in half, and put together again after completion of the construction.
558.
VI, 33.
559.
∠ABC = 2∠ADC (III, 20); ZH = 2ZT by construction.
560.
The polygon sought for is built up from congruent isosceles triangles in which the angles at the center of the circle stand in a given ratio to the full angle. For a polygon with n sides, we get 2π/nfor the vertex angle, and (π − 2π/n)/2 for the angles at the base. The ratio will be 4:(n − 2) in modern notation.
561.
In contrast to Props. 35–38, the propositions of this group do not arise from a generalization of plane or solid problems. They are in principle beyond the reach of plane and solid geometry, because they involve the determination of a ratio between a circular arc and a straight line (π).
562.
Prop. 26, Addition.
563.
Circumferences have the ratio of diameters, or of radii (XII, 2, Circ. mens. 1, VI, 1). A similar proposition was already used in Props. 26, 30, and 36 and will be used again in Prop. 40. For details see the section * in the proof protocol of Prop. 26.
564.
Data 1 with Prop. 26, Addition.
565.
Data 2 with V, 16.
566.
Data, def. 5.
567.
Choose a circle b, with radius r, rectify it by means of the quadratrix into a straight line d. Determine r’ with r:r’ = c:d (VI, 9), and describe the circle a with radius r’. Then circumference a:circumference b = r:r’ = c:d, and since circumference b = d, we get: circumference a = c. Cf. Co p. 108/109, F.
568.
Only the situation where the arc is at most a semicircle is envisaged. Thus, the given ratio in Prop. 40 is not arbitrary. It is also, necessarily, larger than 1:1 (in modern terms).
569.
Analysis-assumption; without loss of generality, C is chosen as the midpoint of arc ACB.
570.
βεβηκυῖα, a hapax legomenon in Coll.IV.
571.
∠EHL = ∠AXC, where X is the midpoint of the sought circle. LM corresponds to ½ AB, i.e., to AR.
572.
ἰδίωμα (where one might have expected σúμπτωμα).
573.
Arc ZE:arc LE = ZH:TN (symptoma); arc ZE:ZH = arc LE:TN (V, 16); but arc ZE:ZH = ZH:HK (Prop. 26) = LH:HK; ⇒ arc LE:TN = LH:HK.
574.
ΔHTN ~ ΔHLM on parallels TN and LM; TH:HL = TN:LM (VI, 4); we have just seen: arc LE:TN = LH:HK; thus: TH:HK = arc LE:LM (V, 23).
575.
First, we show that arc AC:AR = arc EL:LM. AR and LM are half-chords under equal angles. Similar arcs are in the ratio of the corresponding radii (this proposition was used in Props. 26, 36, and 39, and a similar one in Prop. 30; see section * in the proof protocol of Prop. 26). arc AC:arc EL = r1:r2. Consider ΔARX ~ ΔLHM ⇒ r1:r2 = AR:LM; thus, arc AC:AR = arc EL:LM (V, 16). Above, it was shown that arc EL:LM = TH:HK. We now get arc AC:AR = TH:HK.
576.
arc AB = 2arc AC, AB = 2AR by construction.
577.
Hultsch prints AB G, Tr 125, 16 prints the mathematically correct AΓB.
578.
HK is given in the sense that one can posit a freely chosen, but fixed quadrant ZHE with an inscribed quadratrix, and in it, K, and therefore HK are uniquely determined. The quadratrix has to be assumed. The quadrant with quadratrix was assumed to be given in position only, not in size, at the outset of the analysis; the actual size of the quadrant with quadratrix is irrelevant for the analysis and synthesis here.
579.
Data 2.
580.
Data, def. 6.
581.
Data 26.
582.
HT (= HL) and HE are given in position. Therefore, the angle between them is clearly given.There is no directly relevant entry in the Data. Hultsch (295, #2/3 Hu) and Ver Eecke (1933b, p. 229, #2) offer a different interpretation for the conclusion of Prop. 40. See the commentary.
583.
CX is the perpendicular bisector of AB, and AB is given; Data 29.
584.
Data 29 (AR is given in position, and the angle RAX is given in magnitude).
585.
X is given (Data 25). With X and A given, so is the circle with center X and radius XA (Data 26); by construction, B lies on it as well.
586.
D on HZ so that HD:HK equals the given ratio (VI, 9).
587.
R is the midpoint of AB. The right angle determines the position of RX, whereas the point X is as yet not determined in position.
588.
The easiest way to do this is by constructing a triangle congruent to ΔNHT, or ΔMHL with one side on AB, ∠A = ∠NTH, and producing (if necessary) the sides around A to meet XC in R, and X.
589.
The apodeixis is not given by Pappus. It is easily reconstructed from the analysis. See the commentary.
590.
Incommensurable lines were treated in Props. 2 and 3.
591.
For the construction of lines incommensurable in length cf. X, 10 ff., e.g., X, 11.
592.
Arc AC:arc MC = AB:EK and arc AC:arc LC = AB:DN (symptoma); EK:DN = arc MC:arc LC (V, 16/22). arc MC:arc LC = ∠EBZ:∠DBZ (VI, 33).
593.
EK:DN = HB:BT by construction, and these lines are incommensurable by construction.
594.
Cf. the above meta-theoretical passage before Prop. 31, where it is reported that Archimedes was Pappus is going to provide an analysis to show that Archimedes’ neusis can be determined as the intersection of a parabola and a hyperbola. The neusis in Props. 42–44 is closest to SL 9, but an analogous argument could be given for SL 7 and 8. The hyperbola (Prop. 42) and the parabola (Prop. 43) are considered as solid loci. See the commentary on the use, the power, and the limits of geometrical analysis for the determination of the “degree” of a problem.
595.
This is an indication that there may very well have been some move, on the part of ancient geom eters, toward a standardization of “solid” problems via reduction to typical neusis with standard constructions.
596.
As in the case of Prop. 40, this ratio is not completely arbitrary. For the upcoming analysis to work, we need CD ≥ DE. In Prop. 44, we will need the equivalent to CD = DE. Note the analogy to Prop. 34a for the starting point of the argument in Prop. 42.
597.
Z is the point of intersection with AB. It is given (Data 25, Data 28).
598.
Because C — K — Z is assumed, we must have CD ≥ DE, as noted above.
599.
CZ is given in position and length (Data 26); CD:DE is given by hypothesis. ZH and ZK are given in length (Data 2). They are also given in position ⇒ T and K are given (Data 27).
600.
CZ:ZT = CD:DE by construction. CD:DE is given ⇒ CZ²:ZT² = CD²:DE², and this ratio is givenas well (Data 50).
601.
The above proportion implies (CD² − CZ²):(ED² − ZT²) = CZ²:ZT², so both ratios are given. We now show that CD² − CZ² = ZD² (I, 47) = EH², and that (ED² − ZT²) = KH × HT. Then EH²: KH × HT is given. ED² = ZH² = ZT² + TH² + 2ZT × TH (II, 4). 2ZT × TH = KT × TH (construction, VI, 1). TH² + KT × TH = KH × TH (II, 3). So ED² = ZT² + KH × HT, and KH × HT = ED² − ZT².
602.
The converse of Con. I, 21 is used in analysis. According to Con. I, 21, all points on the hyperbola through T with diameter TK, a latus rectum t with t:HK = EH²:TH × KH, and ordinates parallel to AB fulfill the above equation. For the analysis, we can “conclude”: E lies on this hyperbola; cf. Prop. 34a.
603.
Such a rectangle can be constructed using II, 14.
604.
EZ is given in length (Data 57). Because E is given (Data 7 and Data 27), EZ is also given in position (Data 29), and so Z is given (Data 27).
605.
EC² = DH² by construction. Let t be the given line. EB² = t × EZ by construction. However, EB² = AC × CB + EC² (II, 5). By hypothesis, AC × CB = t × DC, and so EC² = EB² − ² AC × CB = t × EZ − t × CD = t × ZH.
606.
Converse of Con. I, 20. Con. I, 20 shows that all points on a parabola with vertex Z, diameter EZ, parameter t and ordinates parallel to AB fulfill the equation. The analysis can use the converse, even if it is not a theorem.
607.
Prop. 44 is not included in Co. Commandino even suggests that Coll. IV ends after Prop. 43. The transmitted manuscripts have no figure for Prop. 44. The figure given by Hultsch p. 303 is badly misleading and was not used here. Hultsch himself supplied a correction in the appendix of his edition, pp. 1231–1233. This passage in the appendix also contains a helpful explanation of Props. 42–44 by Baltzer.
608.
Translating the reading in Tr 127, 21/22 for a lacuna in Hu 300, 21/22.
609.
Both the main manuscript A and Hu have “C” here. Hultsch corrected his reading in the appendix to his edition p. 1232/1233. Tr also prints the mathematically correct “A.” See the apparatus to the Greek text.
610.
Analysis-assumption.
611.
Z does not necessarily lie on the circumference of the circle.
612.
For the lacuna at 302, 8 Hu, Tr 128, 3 prints ἡ AΔ τò Z σημεῖoν αρα, the resulting meaning coincides with the paraphrase given above, and with Hultsch’s conjecture ad locum.
613.
III, 35.
614.
AD = ZD by construction. Prop. 42 states that Z lies on a (uniquely determined) hyperbola.
615.
DE is given in the problem.
616.
Prop. 43.
617.
Z is the point of intersection of the parabola and the hyperbola.
618.
For this lacuna at 302, 12 Hu, Tr 128, 7/8 prints: αναλúεταƖ αρα. Toúτω (τῴ πρoβλήματƖ). Therefore, it is subjected to analysis. This problem.
619.
SL 18 uses SL 7/8.
620.
For the lacuna at 302, 15 Hu, Tr 128, 11 prints: δυνατóν γὰρ ωϛ ἀπo(δείκνoυσƖν) (For it is possible, as they show).This makes Hu’s addition of ωϛ and ἒστƖν in the following line unnecessary.
621.
Unfortunately, no such argument survives. For reconstructions and bibliographical references see the commentary.

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Sefrin-Weis, H. (2010). Annotated Translation of Collectio IV. In: Sefrin-Weis, H. (eds) Pappus of Alexandria: Book 4 of the Collection . Sources and Studies in the History of Mathematics and Physical Sciences. Springer, London. https://doi.org/10.1007/978-1-84996-005-2_2

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