Apollonius of Perga’s on Conics: Book Eight Restored or the Book on Determinate Problems Conjectured

Fried, Michael N.

doi:10.1007/978-1-4614-0146-9_7

Michael N. Fried²

Part of the book series: Sources and Studies in the History of Mathematics and Physical Sciences ((SHMP))

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Abstract

When I had set out to publish the [Conics] of Apollonius, it disturbed us not to a small degree that in the Arabic Codex the last book of the Conics was missing. But you, as it is by [your] happy nature, felt at once that there is no need to deplore [the loss], but that surely to some degree it could be restored, given the fact that in Pappus’ Mathematical Collection, [Pappus] himself passed on lemmas serving [what] was to be demonstrated in the seventh and the eighth book of the Conics at the same time; whereas in the other books, different [lemmas] to the different [books] are provided.

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Notes

1.
Henry Aldrich (1648–1710) was dean of Christ Church College, Oxford from 1689 until his death in 1710. It was Aldrich who persuaded Halley to collaborate with David Gregory in preparing an edition of the works of Apollonius. See the introduction above and Suttle (1940) for more details about Aldrich. The letters S.P. stand for salutem plurimam, “many greetings.” It is a standard Latin epistolary formula: I decided to leave this untranslated to emphasize its formulaic character.
2.
This, indeed, is how Apollonius refers to Book VII in the letter introducing Book I. However,diorisms or limits of possibility, limits within which a given problem can be solved, are not referred to as the main topic of Book VII in the letter introducing Book VII. There, Apolloniusdescribes the book as follows: “In this book [Book VII] are many wonderful and beautiful things on the topic of diameters and the figures constructed on them, set out in detail. All of this is of great use in many types of problems, and there is much need for it in the kind of problems which occur in conic sections which we mentioned, among those which will be discussed and proven the eighth book of this treatise.” (Here, and throughout, translations from Book VII are from Toomer (1990). Also, unless indicated otherwise (i.e., by my initials, MNF), bracketed remarks in quotations from Toomer are Toomer’s own.)
3.
Every proposition in the reconstruction.
4.
Conics VII.5 states: “If there is a parabola, and one of its diameters is drawn in it, and from the vertex of that diameter a perpendicular is drawn to the axis, then the parameter of the lines drawn from the section to the diameter parallel to the tangent drawn from the vertex of the diameter, [i.e., when figures are applied to that line, [those ordinates] are equal in square to them—and that line is its [the diameter’s] latus rectum—is equal to the latus rectum of the axis increased by four times the amount cut off from it by the perpendicular from the axis adjacent to the vertex of the section.”
5.
Let the intersection of BZ and KH be X. Then, by VII.5, lr.(ΛN)=4XM + lr.(axis) \(= 4XM + 4EB = 4XM + 4XK = 4KM\) where I am using “lr.(ΛN)” as an abbreviation for “the latus rectum of diameter ΛN”, “lr.(axis)” for “the latus rectum of the axis,” and so on. Where there is no risk of confusion, I shall just write “lr” for the latus rectum in question.
6.
There is no separate diagram for proposition II; Halley is still referring to the diagram for the previous proposition. See the remarks on the diagrams in the introduction.
7.
VII.32 states the diorism: “[In] every parabola, the latus rectum which is the parameter of the ordinates falling on the axis is the smallest of the latera recta which are the parameters of the ordinates falling on the other diameters; and the latus rectum which is the parameter of the ordinates falling on [one of] those diameters closer to the axis is less than the latus rectum which is the parameter of the ordinates falling on [a diameter] farther [from it].”
8.
Elements IV.5: “About a given triangle to circumscribe a circle.” Unless noted otherwise, I am taking advantage of Heath’s (1926) translations.
9.
Halley’s use of “semi-latis rectum” in the opening of the proposition and “half of the latus rectum” here is an example of the sort of inconsistency that I described in the introduction. I reemphasize here that I try to preserve Halley’s own formulations, including such inconsistencies, as far as possible throughout the translation.
10.
This is the enunciation, essentially, of VII.29.
11.
Conics VII.29 tells us that if D and D′ are any two diameters of a hyperbola, and L and L′ are their respective latera recta, then sq.D-rect.D,L=sq.D′-rect.D′,L′. Hence, rect.D,(D-L)=rect.D′,(D′-L′), from which it also follows that rect.1/2D,(1/2D-1/2L)=rect.1/2D′,(1/2D′-1/2L′).
12.
That is, the rectangle contained by the segments, BA and AZ. Halley, in this, is following Apollonius’s and Euclid’s usual notational practices.
13.
Elements III.35 (which Halley needs for the lower diagram) states: “If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other”; Elements III.36 states: “If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut the circle and the other touch it, the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference will be equal to the square on the tangent.” It follows immediately (and it is this result, which is not stated explicitly in Euclid, that Halley needs for the upper diagram) that if two straight lines in a circle cut one another outside the circle the rectangle formed by the whole of the one straight line which cuts the circle and the segment of it intercepted on it outside between the point of intersection and the convex circumference is equal to the rectangle formed by the same segments of the other straight line.
14.
Again, this is the enunciation of VII.30.
15.
The reasoning is the same as in the previous proposition, namely, Conics VII.30 tells us that if D and D′ are any two diameters of an ellipse, and L and L′ are their respective latera recta, then sq.D+rect.D,L=sq.D′+rect.D′,L′. Hence, rect.D,(D+L)=rect. D′,(D′+L′), from which it also follows that rect.1/2D,(1/2D+1/2L)=rect.1/2D′,(1/2D′+1/2L′).
16.
By “the figure of the axis,” Halley means the figure with respect to the axis, that is, the rectangle whose sides are the axis and the latus rectum with respect to the axis (for this reason the latus rectum is called the latus rectum, the upright side—it is the upright side of that rectangle!).
17.
Halley means by this that a segment equal to the latus rectum is drawn from these points to A, so that, ΔA = Aδ.
18.
Proposition VII.2 is cited only because in the course of it the homologue is first introduced. The complete proposition runs as follows: “If the axis in a hyperbola is extended in a straight line so that the part of it falling outside of the section is the transverse diameter, and a line is cut off adjoining one of the ends of the transverse diameter such that the transverse diameter is divided into two parts in the ratio of the transverse diameter to the latusrectum, while the line cut off corresponds to the latusrectum, and a line is drawn from that end of the transverse diameter which is the end of the line which was cut off to the section, in any position, and, from the place where [that line] meets it [the section], a perpendicular is drawn to the axis, then the ratio of the square on the line drawn from the end of the transverse diameter to the rectangle contained by the two lines between the foot of the perpendicular and the two ends of the line which was cut off is equal to the ratio of the transverse diameter to the excess of it [the transverse diameter] over the line which was cut off.
“And let the line which was cut off be called the ‘homologue’.”
19.
It is worthwhile to review at this point some of the conventional words used to describe the manipulations of proportions. Let a:b::c:d. Then the proportion obtained componendo is a+b:b::c+d:d; the proportion obtained separando is a-b:b::c-d:d; the proportion obtained convertendo (Halley typically writes, simply, “by conversion,” per conversionem) is a:a-b::c:c-d; the proportion obtained, permutando or alternando, is a:c::b:d (provided a,b,c,d are magnitudes of the same kind). If a:b::c:d and b:e::d:f, then the proportion obtained ex aequali is a:e::c:f. These relationships and others are described by Euclid in the definitions opening Book V of the Elements.
20.
Remember AN = ΓΞ and AΔ=Aδ. So, with that in mind, we have from NA:ΓN::ΞΓ: EA::AΔ:AΓ, NA:NΞ::ΞΓ:AΞ − ΞΓ::AΔ:ΓA − AΔ::AΔ:Γδ.
21.
It is not necessary to apply both VII.29 and VII.13. For VII.13 states that, “[In] every hyperbola, the difference between the squares on its axes is equal to the difference between the squares on any pair of its other conjugate diameters whatever.” So, if D and d are conjugate diameters, while X and x are the conjugate axes, then VII.13 tells us that sq.D-sq.d=sq.X-sq.x. But, by the definitions following I.16 and proposition I.60, we have sq.x=rect.X,lr.X (where “lr.X” indicates the latus rectum of the figure on X—a notation I shall continue to use for convenience). Therefore, sq.D-sq.d=sq.X-sq.x=sq.X-rect.X,lr.X. Alternatively, sq.D-sq.d=sq.D-rect.D,lr.D by I.60, while sq.D-rect.D,lr.D=sq.X-rect.X,lr.X by VII.29; therefore, again, sq.D-sq.d=sq.X-rect.X,lr.X.
22.
VII.6 states: “If there are constructed on the extension of the axis of a hyperbola two lines adjacent to the two ends of the axis which is the transverse diameter, each of them being equal to the line which we called ‘homologue’, and placed as it [the homologue] is placed, and two conjugate diameters from among the diameters of the section are drawn, and, from the vertex of the section, a line is drawn parallel to the ‘erect’ diameter of the two to cut the section, and from the place where it intersects it a perpendicular is drawn to the axis: then the ratio of the transverse of the two conjugate diameters to the erect one is equal in square to the ratio of the line between the foot of the perpendicular and the end of the more remote of the two homologues to the line between the foot of the perpendicular and the end of the nearer of the two homologues; and the ratio of the transverse diameter to the parameter of the lines drawn to it [the transverse diameter] parallel to the second diameter, which [parameter] is its latus rectum, is, in length, equal to the ratio of the two lines which we mentioned previously to each other in length.” The “specification” is precisely what Halley goes on to say.
23.
Despite Halley’s general care in maintaining Apollonius’s style, this is a manifestly un-Apollonius manipulation, and it is clearly algebraically motivated. He argues as follows: sq.BK:rect.NΞ,ΓΔ::ΞM:NΞ, therefore—and now I shall shift to an algebraic notation (even though Halley wants to say rect.NΞ,(sq.BK)!)—NΞ ⋅BK ²=ΞM ⋅(NΞ ⋅ΓΔ)=NΞ ⋅(ΞM ⋅ΓΔ), from which we have BK ²=ΞM ⋅ΓΔ.
24.
As before, rect. AMΓ means rect.AM,MΓ.
25.
In fact, it is the converse of I.21. I.21 states: “If in an hyperbola or ellipse or in the circumference of a circle straight lines are dropped ordinatewise to the diameter, the squares on them will be to the areas contained by the straight lines cut off by them beginning from the ends of the transverse side of the figure, as the upright side of the figure is to the transverse, and to each other as the areas contained by the straight lines cut off (abscissas), as we have said.” For quotations from Books I–III, I shall take advantage of Taliaferro’s translations (as revised by Dana Densmore in the Green Lion edition).
26.
Halley’s use of the verb posse here, and in similar passages, is exactly parallel to the use of the Greek dynasthai, which, in Greek mathematical works, means “equal in square” (see the remarks in the introduction).
27.
And by I.60 or the definition of the second diameter, as described above.
28.
ΠO is chosen to be the mean proportional between AΓ and AΓ-lr.AΓ. Therefore, sq.ΠO=rect.AΓ,(AΓ-lr.AΓ)=sq.AΓ-rect.AΓ,lr.AΓ. Let D be the given diameter and d the diameter conjugate to it. Then, by VII.29, sq.ΠO=sq.AΓ-rect.AΓ,lr.AΓ=sq.D-rect.D,lr.D. But, by I.60, rect.D,lr.D=sq.d. Therefore, sq.ΠO=sq.D-sq.d or sq.d=sq.D-sq.ΠO. Thus, setting OP equal to the given diameter, D, PR will, by the Pythagorean theorem, be equal to d. As for the diorism at the end, it follows from VII.29, and Apollonius states this explicitly in the corollaries following VII.31, that if the axis of a hyperbola is greater than its latus rectum, then every diameter is greater than its latus rectum.
29.
That the axis is the shortest transverse diameter of a hyperbola is not proven explicitly by Apollonius, but it is stated in VII.28 and follows from V.34 (see Toomer, 1990, p. 603, note 107).
30.
As in the last proposition, VII.3 is cited here only because it introduces the homologue for the ellipse. The proposition itself states: “If a line is constructed on the extension of one of the axes of an ellipse, whichever axis it may be, and one of [the line’s] ends is one of the ends of the transverse diameter, while the other end is outside of the section, and the ratio of [the line] to the line between its other end and the remaining end of the [transverse] diameter, and a line is drawn from the end common to the [transverse] diameter and the line constructed on the axis, to any point on the section, and from the place where it meets [the section] a perpendicular is drawn to the axis, then the ratio of the square on the line which was drawn [to the section] to the rectangle contained by the two lines between the foot of the perpendicular and the two ends of the [first] line which was constructed on the axis is equal to the ratio of the transverse diameter to the line between those two ends of the transverse diameter and the line which was constructed that are different from each other. Let the line which was constructed be called the ‘homologue’.”
31.
This is the same as separando, defined above.
32.
By definition of the homologue, ΞΓ:AΞ : : AΔ : AΓ : : lr. (Axis) : Axis. So componendo, ΞΓ: ΞΓ + AΞ::AΔ:ΔA + AΓ::lr. (Axis): Axis + lr. (Axis).
33.
Proposition VII.12 is the proposition for the ellipse corresponding to VII.13, which we have seen, for the hyperbola: “[In] any ellipse, the sum of the squares on any two of its conjugate diameters whatever is equal to the sum of the squares on its two axes.” So, if D and d are conjugate diameters and X and x are the conjugate axes, VII.12 tells us that sq.D+sq.d=sq.X+sq.x. But, by I.15, sq.x=rect.X,lr.X; therefore, sq.D+sq.d=sq.X+rect.X,lr.X. Alternatively, by I.15, sq.D+sq.d=sq.D+rect.D,lr.D. But by VII.30 sq.D+rect.D,lr.D=sq.X+rect.X,lr.X. So, again VII.12 or VII.30 is needed, but not both.
34.
Again, this is the proposition for the ellipse analogous to VII.6 for the hyperbola. VII.7, in full, states: “If there are constructed on the extension of the axis of an ellipse two lines at the two ends of [the axis], each of them being equal to the homologue, and two conjugate diameters are drawn in the section, and, from the vertex of the section, a line is drawn parallel to one of the conjugate diameters so as to meet the section [again], and, from the place where it meets [the section], a perpendicular is drawn to the axis: then the ratio of the diameter which is not parallel to the line which was drawn, to the other diameter, is equal in square to the ratio to each other of the two parts (of the line between the ends of the two homologues which are not the ends of the diameter) into which it is cut by the perpendicular: according to how the two homologues are placed: if [they are found] on the major axis, they are outside the section, and if on the minor axis, then they are on the axis itself.
“And the ratio of the diameter mentioned to the parameter of the ordinates falling on it (which are parallel to the other [conjugate] diameter) is [also] equal to the ratio mentioned.”
35.
The quasi-algebraic reasoning here is similar to that at the end of the analysis in proposition V.
36.
That is, to ΛΓ and ΛA, respectively.
37.
See problem V above.
38.
Since, by definition, sq.ΠΓ=sq.BK+sq.ZH, by construction, AΔ=ΠΓ, AB = BK, and because trgl.ABΔ is right.
39.
In other words, AB:BΔ::BΔ:BΓ or sq.BΔ=rect.AB,BΓ, that is, sq.ZH=rect.BK,BΓ.
40.
If BK were the mean proportional between the major axis, say, and its latus rectum, then the problem would be reduced merely to finding the minor axis.
41.
Proposition VII.21 states: “If there is a hyperbola, and its transverse axis is greater than its erect axis, then the transverse diameter of each pair of conjugate diameters among its other diameters is greater than the erect diameter of [that pair]; and the ratio of the greater axis to the lesser axis is greater than the ratio of the transverse diameter to the erect diameter among the other conjugate diameters; and the ratio of a transverse diameter nearer to the greater axis to the erect diameter conjugate with it is greater than the ratio of a transverse diameter farther [from that axis] to the erect diameter conjugate with it.”
42.
It should be kept in mind that since PΣ :ΣT ::ΣT :ΣY, we have PΣ:ΣY ::sq.PΣ:sq.ΣT. Thus, the fact that PΣ:ΣY is given means that the ratio of the squares on the diameters is given.
43.
Because PΣ = Σϕ, ΣY are given, also \(\Sigma \phi - \Sigma Y = \phi Y\) and ΣP are given; therefore, the ratio ϕY :ΣP is given.
44.
In other words, sq.PΣ-sq.ΣT:sq.PΣ; in referring to PΣ as, literally, “the line analogous to BK,” I understand Halley to be referring to the positions of PΣ and BK in the proportion PΣ:ΣT::BK:ZH.
45.
Which is obviously true.
46.
Proposition VII.22 states: “If there is a hyperbola, and its transverse axis is shorter than its erect axis, then the transverse diameter of each pair of diameters among the other conjugate diameters is shorter than the erect diameters of [that pair]; and the ratio of the shorter axis to the longer axis is less than the ratio of any of the other transverse diameters to the erect diameter conjugate with it; and the ratio of a transverse diameter nearer to the shorter axis to the erect diameter conjugate with it is less than the ratio of [a transverse diameter] farther [from that axis] to the diameter conjugate with it.”
47.
Halley’s statement is essentially that of VII.23, namely, “If the two axes of a hyperbola are equal, then every conjugate pair of its diameters is equal.”
48.
That is, in the ratio that an equal has to an equal.
49.
As in the synthesis in proposition VI.
50.
Proposition VII.24 states: “If there is an ellipse, and conjugate diameters are drawn in it, then the ratio of the greater of each pair of conjugate diameters to the lesser is less than the ratio of the major axis to the minor axis: and, for any two pairs of conjugate diameters, the ratio of the greater diameter which is nearer to the major axis [than the other greater diameter] to the lesser [diameter] conjugate with it is greater than the ratio of the greater diameter which is farther from the major axis to the lesser diameter conjugate with it.” The second condition stated by Halley, namely, that the ratio of the conjugate diameters be not less than the ratio of the minor to major axis, follows immediately from the second part of VII.24.
51.
Since AO is the hypotenuse of the right triangle whose legs are the half-axes AΘ and ΘO.
52.
Proposition VII.8 refers to the diagrams in VII.6, 7, which are lettered in precisely the same way as in propositions V and VI in Halley’s text, and, with that, it states that “…the ratio of the square on AΓ (which is the transverse diameter) to the square on BK and ZH, the two conjugate [diameters], when they are joined together in a straight line, is equal to the ratio of (NΓ ⋅MΞ) [Toomer’s notation] to the square on a line equal to line MΞ plus the line which is equal in square to (MN ⋅MΞ) [again, Toomer’s notation].” Thus, if P is the mean proportional between MN and MΞ, that is, if sq.P=rect.MN, MΞ, then sq.AΓ:sq.(BK+ZH)::rect. NΓ,MΞ:sq.(MΞ + P).
53.
The argument is as follows. With P defined as in the last note, sq.AΓ: rect. NΓ,MΞ:: sq.(BK + ZH):sq.(MΞ+P). But sq.AΓ=rect.NΓ,ΓΔ. Therefore, sq.AΓ: rect. NΓ,MΞ:: rect.NΓ,ΓΔ: rect. NΓ,MΞ, from which we have (by Elements. II.4 and the definition of P), ΓΔ:MΞ::sq.(BK + ZH):sq.(MΞ + P):: sq.(BK + ZH):sq.MΞ+sq.P+2rect.MΞ, P::sq.(BK + ZH):sq.MΞ+rect.MΞ,MN+2rect.MΞ,P. So, ΓΔ:MΞ::sq.(BK + ZH):sq.MΞ + rect.MΞ, MN+ 2rect.MΞ, P or ΓΔ:MΞ::rect.ΠP,(BK + ZH):rect.\(M\Xi (M\Xi + MN + 2P)\). The next step, I suspect, was worked out by Halley with the help of algebra (we have seen evidence for that before in the context of proposition V), for by saying MΞ is “found on both sides” Halley seems to justify the step by “canceling” the common term; however, it can be given a classical justification: MΞ:ΠP::rect.MΞ(\(M\Xi + MN + 2P\)): rect.ΠP(\(M\Xi + MN + 2P\)); therefore, since ΓΔ:MΞ::rect.ΠP, (BK + ZH):rect.MΞ(\(M\Xi + MN + 2P)\), we have, ex aequali, ΓΔ:ΠP:: rect.ΠP, (BK + ZH):rect.ΠP,(\(M\Xi + MN + P) :: BK + ZH : (M\Xi + MN + 2P)\), or \(\Gamma \Delta : \Pi P :: BK + ZH : (M\Xi + MN + 2P)\). But \(M\Xi + MN = MN + N\Theta + \Theta \Xi + MN\), and NΘ = ΘΞ, so that \(M\Xi + MN = 2(MN + N\Theta ) = 2M\Theta \). Therefore, \(\Gamma \Delta : \Pi P :: BK + ZH : 2(M\Theta + P) :: 1/2(BK + ZH) : (M\Theta + P)\).
54.
Namely, the line MΘ + P.
55.
Halley has in mind the equality \(\Theta O = OM + M\Theta = M\Theta + P\); thus, removing MΘ from both side, we obtain, OM = P.
56.
For in this case the two axes and their latera recta are all equal to one another, so the homologues are equal to the semi-axes.
57.
Elements II.6 states: “If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line.” So if the half is equal to diameter ZH and the half and the added line is equal to BK, the conjugate to ZH, then the whole line is BK + ZH and the added line is BK − ZH. Elements II.6 tells us, then, that the rectangle contained by BK + ZH and BK − ZH, together with the square on the half, that is, on ZH equals the square on the square on the half and the added line, that is, on BK. Hence, \(\mathrm{rect}.(BK + ZH), (BK - ZH) + \mathrm{sq}.ZH = \mathrm{sq}.BK\), or \(\mathrm{rect}.(BK + ZH), (BK - ZH) + \mathrm{sq}.ZH = \mathrm{sq}.BK -\mathrm{sq}.ZH\).
58.
That is, so that ΓΔ : BK : : BK : MΞ or ΓΔ : ZH : : ZH : NM.
59.
For above, and in the previous problems, M is found first, and then, having found it, BK and ZH are found.
60.
Proposition VII.25 states: “[In] every hyperbola, the line equal to [the sum of] its two axes is less than the line equal to [the sum of] any other pair whatever of its conjugate diameters; and the line equal to the sum of a transverse diameter closer to the greater axis plus its conjugate diameter is less than the line equal to the sum of a transverse diameter farther from the greater axis plus its conjugate diameter.”
61.
The argument beginning here is, mutatis mutandis, follows along the lines of the corresponding argument in Proposition IX.
62.
Halley must mean Elements II.5, which states: “If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.” In this case, the whole line is NΞ, which is divided into equal parts by the center of the ellipse Θ and into unequal parts by the point M; therefore, the rectangle contained by NM, MΞ together with the square on MΘ is equal to the square on ΘΞ, that is, the rectangle contained by NM,MΞ is equal to the difference between the square on ΘΞ and that on MΘ.
63.
Rect.NM,MΞ=sq.ΘΞ-sq.MΘ or sq.MΘ=sq.ΘΞ-rect.NM,MΞ. But rect.NM,MΞ=sq.ΞK. Therefore, sq.MΘ=sq.ΘΞ- sq.ΞK.
64.
The arc will also cut the axis at another point. This point, μ, is not referred to in the demonstration but shown in the diagram.
65.
Elements II.7 states: “If a straight line be cut at random, the square on the whole and that on one of the segments both together are equal to twice the rectangle contained by the whole and the said segment and the square on the remaining segment.” The square on the remaining segment is, therefore, the square on the difference between the whole and the first segment, so that, if we take the whole to be a diameter and one of the segments to be its conjugate, then the sum of the squares on the two diameters will, by Elements II.7, equal the rectangle contained by the conjugate diameters together with the square on the difference of the conjugate diameters.
66.
Proposition VII.26 states: “[In] every ellipse, the sum of its two axes is less than [the sum] of any conjugate pair of its diameters; and the sum of any conjugate pair of its diameters which is closer to the two axes is less than the sum of any conjugate pair of its diameters farther from the two axes; and the sum of the conjugate pair of its diameters each of which is equal to the other is greater than that of any [other] conjugate pair of its diameters.” Halley’s last remark follows from this propostion and VII.12. For if D and d are conjugate diameter, E is one of the equal conjugate diameters, and X and x are the axes, VII.26 tells us that D+d ≤ 2E or sq.(D+d) ≤ 4sq.E. But, by VII.12, 4sq.E=2(sq.E+sq.E)=2(sq.X+sq.x). Therefore, for any pair of conjugate diameters, D and d, we have sq.(D+d) ≤ 2(sq.X+sq.x).
67.
As in proposition IX.
68.
Proposition VII.9 tells us that with P defined as above, namely, sq.P=rect.MN, MΞ, we have sq.AΓ:sq.(BK-ZH)::rect.NΓ,MΞ: sq.(MΞ-P). With this, or rather, sq.AΓ:rect.NΓ,MΞ::sq.(BK-ZH):sq.(MΞ-P), the rest, as Halley points out, follows just as in proposition IX. The subsequent claims also, mutatis mutandis, are proven as in proposition IX.
69.
“Twice PΘ” because NΘ = ΘΞ, so that \(P\Xi -\mathrm{NP} = (P\Theta + \Theta \Xi ) - (N\Theta - P\Theta ) = (P\Theta + \Theta \Xi ) - (\Theta \Xi - P\Theta ) = 2P\Theta \).
70.
In Greek mathematics, “analogy,” analogia, is the term for “proportion.” Halley uses the Greek word (which indeed appears in Greek in Halley’s text) in this way, not only here, but also in problems XX, XXIV, XXVI, XXXII, and XXXIII.
71.
Hinc modo satis expedito tam nonum quam hoc undecimum problema contruxeris. It is curious that Halley does not provide this procedure in the context of proposition IX—the sentence almost has the feel of an afterthought.
72.
Since ΣΦ is a perpendicular bisector of PY.
73.
By the Pythagorean theorem, sq.ΣY -sq.ΣT=sq.ΣP-sq.ΣT=sq.TY =sq.Axis-rect.Axis, lr.Axis. But sq.Axis-rect.Axis, lr.Axis=rect.(D-d),(D+d), where D and d are conjugate diameters. Furthermore, sq.ΣP-sq.ΣT=rect.(ΣP-ΣT),(ΣP+ΣT)=rect.TR,(ΣP+ΣT)=rect.(D-d),(ΣP+ΣT). Therefore, ΣP+ΣT=D+d. And, since also ΣP-ΣT=D-d, it follows that ΣP=D and ΣT=d.
74.
“[In] every ellipse, or hyperbola in which the two axes are unequal, the increment of the greater axis over the lesser is greater than the increment of [the greater of] any conjugate pair [sic] of its diameters over the diameter conjugate with it; and the increment of [the greater of a pair of] them nearer to the greater axis over the diameter conjugate with it is greater than the increment of [the greater of a pair of them] farther [from the major axis] over the diameter conjugate with it.”
75.
This already follows from VII.13 (or VII.29). Halley probably wants to bring out that the construction given at the end of VIII.11 does not refer to any particular hyperbola and, therefore, to all hyperbolas.
76.
Previous proposition, first paragraph.
77.
See the previous proposition with PΘ playing the part of OΘ and ΘM playing the part of ΘΞ.
78.
Again with P defined so that sq.P=rect.MN,MΞ, we have OΘ=ΘΞ-P or OΞ=ΘΞ-OΘ = P; so, sq.OΞ=rect.NM,MΞ.
79.
See note to proposition X above.
80.
The “Pythagorean Theorem.”
81.
That is, to the sum of the squares of any pair of conjugate semi-diameters, or four times the sum of the squares of the conjugate diameters themselves.
82.
Elements III.20 states: “In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.” In this case, since angle AΔB subtending the arc AΔB is three right angles, the angle at the circumference, AZB is half of three right angles.
83.
Elements III.31 states: “In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; and further the angle of the greater segment [i.e., the angle between one side of the angle and the arc of the circle] is greater than a right angle, and the angle of the less segment less than a right angle.”
84.
The line set out equal to the given difference.
85.
For if the sum of the squares on the axes of the ellipse is K ², then the locus of points H, such that \(A{H}^{2} + H{B}^{2} = {K}^{2}\) is the semicircle AHB (in fact, the entire circle—but it suffices, by symmetry, to consider only the semicircle). To each point H, however, there corresponds a unique point Z on the arc AZB, and, therefore, a unique length BZ. Hence, given BZ, H is uniquely determined, and, therefore, also AH and HB are uniquely determined.
86.
Since angle AΓB, at the center, is right.
87.
If the sum of the squares of the axes is given, say, equal to AB ², the conjugate diameters must be pairs of lines such as AH, HB, where H is a point on the semicircle AHB having diameter AB. So, regardless of the latus rectums of the ellipses, if the sums of squares of the axes in two ellipses are both AB ² and a diameter in one is equal to a diameter in the other—equal, say, to BH—then the conjugates of each must be equal to AH.
88.
See appendix 1.
89.
This is the same as Apollonius’s enunciation of VII.10, which is also given in terms of the previous figure, that is, there is no additional protasis.
90.
Why Halley cites VIII.7 here is unclear. Indeed, the fact that rect.NΓ,(axis+latus rectum)=sq.AΓ (=sq.axis) follows from the definition of the homologue: AN : NΓ::latus rectum: axis; hence, (AN + NΓ):NΓ::(axis+latus rectum): axis or AΓ:NΓ::(axis+latus rectum):AΓ, from which the result is immediate.
91.
The words “sublato utrinqueNΓ” here, as in Prop. IX, betray Halley’s use of algebra to work out some of his results. In this case, the algebra must have been something like this: sq.AΓ/rect.\((BZ,HZ) = N\Gamma /P\) (sq.P = rect.NMΞ) and rect.NΓ, ΓΔ=sq.AΓ; therefore, rect.(NΓ,ΓΔ)/rect.\((BZ,HZ) = N\Gamma /P\), so that, “eliminating NΓ from both,” we have ΓΔ/rect.\((BZ,HZ) = 1/P\) (an expression impossible to write in an Apollonian context!), from which we obtain rect.BZ, HZ = rect.ΓΔ, P. Apolloniuscould have drawn the same conclusion as follows: since (reverting to the usual signs for ratios and proportions) sq.AΓ:rect.(BZ, HZ)::NΓ : P, also, with ΓΔ as a common side, sq.AΓ:rect.(BZ, HZ)::rect.NΓ,ΓΔ:rect.P, ΓΔ; but rect.NΓ,ΓΔ=sq.AΓ; therefore, sq.AΓ:rect.(BZ, HZ)::sq.AΓ:rect.P, ΓΔ from which we have the result. Halley could have framed this kind of argument easily enough. That he did not, suggests that such quasi-algebraic formulations were not, to his mind, violations of his otherwise historically faithful approach.
92.
This is a technical term in Greek mathematics: toapply an area A to a line m means to find a line n so that area A=rect.m,n. In the Elements, the idea of application of areas first appears in I.44; in the Conics, the three basic kinds of application of areas—parabolic, elliptic, and hyperbolic—are connected to the characteristic properties, the symptōmata, of the conic sections and account for the names Apollonius gives to the sections.
93.
By the definition of the homologue for the ellipse, NΓ:NA::axis: latus rectum::AΓ: Aδ; therefore, convertendo,NΓ:NΓ-NA::axis: axis − latus rectum, or NΓ:AΓ::AΓ:Γδ, so that rect.NΓ, Γδ=sq.AΓ.
94.
Again, the argument is algebraic: referring, thus, to products rather than rectangles, we have (NΓ ×Γδ)/\((BK\times ZH) = N\Gamma /P\) (sq.P=rect.NMΞ) or, “since NΓ is found on both sides,” Γδ/(BK ×ZH)=1/P, from which we have Γδ ×P = BK ×ZH.
95.
Namely, the rectangle contained by the conjugate diameters.
96.
See note on proposition X.
97.
That is, the rectangle contained by the conjugate axes.
98.
Conics VII.28 states: “[In] every hyperbola or ellipse, the rectangle formed by the product of its two axes is less than the rectangle formed by the product of any conjugate pair whatever of its diameters; and, of the conjugate diameters, for those in which the longer [of the pair] is closer to the greater axis, the product of [that diameter] and the diameter conjugate with it is less than the product of one of those in which it is farther from it [the greater axis] and the diameter conjugate with it.” I ought to mention here that although I am not competent to judge whether “product” is an accurate translation of the Arabic of the Banū Mūsā text, I feel confident to say that it is highly unlikely that this was an accurate translation of the Greek text, which was almost certainly a variation of the phrase ….
99.
This follows, of course, from the second part of VII.28.
100.
Recall the figure is the rectangle contained by the axis and its latus rectum: since ΓΔ equals the axis together with its latus rectum and ΘΓ is half the axis, the rectangle in question equals 1/2(sq.Axis+rect.Axis,latus rectum). Moreover, by Conics I.15, the figure on any diameter equals the square on the conjugate diameter (see note to Prop. VI above); therefore, 1/2(sq.Axis+rect.Axis,latus rectum)=1/2(sq.Axis+sq.conj.Axis).
101.
That is, the sum of BK and HZ.
102.
The argument is as follows. Let the diameters sought be D and d, where D is the larger of the two. First, sq.AB = sq.D+sq.d=sum of the squares on the axes, and sq.BΓ=2rect.D,d, which is given. Now, the point E lies on a circle whose diameter is AB; therefore, triangle AEB is right and sq.AE+sq.BE = sq.AB or sq.AE = sq.AB − sq.BE. But BE = BΓ; therefore, sq.AE = (sq.D+sq.d) − 2rect.D,d=sq.(D − d); therefore, AE = D − d. But sq.AΓ=sq.AB + sq.BΓ=(sq.D+sq.d) + 2rect.D,d=sq.(D + d), so that AΓ=D+d. Hence, with \(AZ = AE =\)D − d and AΓ=D + d, AH = 1/2(\(AZ + A\Gamma ) =\)1/2(2D)=D and \(H\Gamma = A\Gamma - AH =\)(D + d) − D = d.
103.
Conics VII.11 states, just as Halley presents it, that sq.AΓ:(sq.BK+sq.ZH)::ΓN:(NM+MΞ). Like proposition VII.10, proposition VII.11 contains no protasis and refers back to the diagrams in VII.6,7, which are similar to Halley’s diagrams here and in the previous proposition.
104.
As we have already seen, the argument, introduced by the phrase “NΓ being on both sides,” is plainly algebraic. Accordingly, writing the claims as products and quotients, we have \(A\Gamma {}^{\mbox{ 2}}/(BK{}^{\mbox{ 2}} + ZH{}^{\mbox{ 2}}) = N\Gamma /(NM + M\Xi \)), or since \(A\Gamma {}^{\mbox{ 2}} = N\Gamma \times \Gamma \Delta,N\Gamma \times \Gamma \Delta /(BK{}^{\mbox{ 2}} + ZH{}^{\mbox{ 2}}) = N\Gamma /(NM + M\Xi )\), or, since NΓ is on both sides, \(\Gamma \Delta /(BK{}^{\mbox{ 2}} + ZH{}^{\mbox{ 2}}) = 1/(NM + M\Xi \)), from which the result follows.
105.
That is, the width of the rectangle applied to ΓΔ which is equal to half the sum of the squares.
106.
This is stated explicitly in the first corollary following Conics VII.31.
107.
This again is very nearly Apollonius’s own enunciation of Conics VII.14.
108.
Namely, from the basic proportion NΓ:NA::AΓ:Aδ we derive equality sq.AΓ=rect.NΓ,Γδ; then, by VII.14 and with Δ indicating the difference of the squares on the conjugate diameters, we write NΓ:2ΘM::sq.AΓ:Δ::rect.NΓ,Γδ: Δ; then “NΓ being on both sides,” we obtain (now writing the expression algebraically as before) 1/2ΘM=Γδ/Δ, so that Δ=rect. Γδ,2ΘM.
109.
By the third corollary after Conics VII.31.
110.
If D and d are conjugate diameters and X and x are the axes of a hyperbola, then Conics VII.13 tells us that sq.D-sq.d=sq.X-sq.x. Suppose the sum of squares sq.D+sq.d is given. Then 1/2(sq.D+sq.d) ± 1/2(sq.X-sq.x) are given. But 1/2(sq.D+sq.d)+1/2(sq.X-sq.x)=1/2 (sq.D+sq.d)+1/2(sq.D-sq.d)=sq.D. Therefore, sq.D and, accordingly, D is given in magnitude. Similarly, 1/2(sq.D+sq.d)-1/2(sq.X-sq.x) =1/2(sq.D+sq.d)-1/2(sq.D-sq.d)=sq.d. So, d is given in magnitude. Conics VII.12 tells us that, for the ellipse, sq.D+sq.d=sq.X+sq.x. Then since, in VIII.16, sq.D-sq.d is given, 1/2(sq.D-sq.d) ± 1/2(sq.X+sq.x)=1/2(sq.D-sq.d)+1/2(sq.D+sq.d) are given, and, again, we obtain D and d for the ellipse.
Implicitly, this whole development is related to Pappus’s lemma 8 for Conics VII-VIII: “Let the sum of the squares on AB and BΓ be given and also the excess of the squares on AB and BΓ: [I say] that each of AB and BΓ is given” (Collection, VII, Hultsch, p. 996). This is one of the few places in Halley’s reconstruction where one can point to a connection between Halley’s development and Pappus’s lemmas. Given the remarks in Halley’s introduction about the importance of these lemmas for his reconstruction, it is curious that Halley does not mention Pappus explicitly here.
111.
Halley must have in mind Conics II.51: “Given a section of a cone, to draw a tangent which with the diameter drawn through the point of contact will contain an angle equal to a given acute angle.” Since the tangent is parallel to the conjugate diameter, Conics II.51 allows us, alternatively, to find a pair of conjugate diameters containing a given angle.
112.
Recall, this is how Halley refers to Book VII in the letter to Henry Aldrich opening Book VIII and how Apollonius, himself, refers to Book VII in the letter introducing Book I of the Conics.
I might also remark that although the word diorism generally appears in Greek in the text, in this spot Halley writes in Greek-inflected Latin, Theoremata dioristica.
113.
Conics VII.31 states: “When a pair of conjugate diameters is drawn in an ellipse or between conjugate opposite sections [i.e., hyperbolas], then the parallelogram bounded by that pair of diameters with angles equal to the angles formed by the diameter at the center is equal to the rectangle bounded by the two axes.”
114.
By Elements I.35, and because the perpendicular, having been dropped from the semi-diameter, is half the altitude of the parallelogram contained by the conjugate diameters.
115.
Let D, d be the conjugate diameters, N the perpendicular dropped from the extremity of D to d. Then rect.N,d:rect.1/2D,d::N:1/2D by Elements VI.1. The latter ratio is given because the angle contained by D and d is given.
116.
Hence, it is angle PXΘ which is equal to the given angle.
117.
Once it is demonstrated that the rectangle contained by ΘX,ΓΔ is equal to that contained by BK, ZH (as Halley goes on to do in the next paragraph), the reasoning can proceed exactly as in VIII.13 above with ΘX, here, playing the part of ΘO: thus, again, sq.ΘM will be shown equal to sq.XΘ+sq.ΘΞ or sq.XΞ.
118.
The argument is the same as that in the second paragraph of the previous proposition.
119.
It is, of course, angle AΘP that has been made equal to the given angle between the conjugates; but since ΠΘ is perpendicular to AΘ and PΠ is perpendicular to PΘ, it follows that angle AΘP=ΘΠP.
120.
This is Conics II.52: “If a straight line touches an ellipse making an angle with the diameter drawn through the point of contact, it is not less than the angle adjacent to the one contained by the straight lines deflected at the middle of the section.” Once again, the reader must keep in mind that the tangent at the vertex of a diameter is parallel to the conjugate diameter, so II.52 says that the angle between the conjugate diameters cannot be less than the said angle.
121.
Recall, ΓΔ is the latus rectum with respect to the axis AΓ.
122.
Halley has in mind the angle within a circle as in the diagram below:
123.
The argument is this: The sum of the squares of the axes, which is equal to the rectangle AΓΔ, by Conics VII.12 is equal to the sum of the squares on conjugate diameters, sq.BK+sq.ZH. Since the ratio ΘH:HT is given (because the angle between the conjugate diameters is given) and since rect.BK,ZH:rect.Axes::ΘH:HT, the rectangle contained by BK, ZH is given. But by Elements II.4,
sq.BK +sq.ZH+2rect.BK,ZH=sq.(BK + ZH),
and, by Elements II.7,
sq.BK +sq.ZH-2rect.BK,ZH = sq.(BK − ZH).
Therefore, BK + ZH and BK − ZH are given and, accordingly, BK and ZH.
124.
This observation seems to be only the trivial one that, with the exception of the axes, every pair of conjugate diameters containing a given angle, or a given area, or possessing any of the other properties treated in the previous theorems, has a matching pair “inclined to the other side” of the axis, that is, the pair obtained by reflecting the first in the minor or major axis.
125.
The exact enunciations of these propositions are as follows. Elements VI.28: “To a given straight line to apply a parallelogram equal to a given rectilineal figure and deficient by a parallelogrammic figure similar to a given one: thus the given rectilineal figure must not be greater than the parallelogram described on the half of the straight line and similar to the defect.” Elements VI.29: “To a given straight line to apply a parallelogram equal to a given rectilineal figure and exceeding by a parallelogrammic figure similar to a given one.” The cases which are of interest to Halley are those where the parallelogram to which the deficiency or excess is similar is a square; for in each of these cases a rectangle is found equal to a given rectangle and having sides whose sum or difference is equal to a given line (which happens to be the line to which the rectangles are applied!). Thus, suppose the given line is L and the given area is A: in the first case, we are looking for lengths a and b (b<L) such that rect.a,b=A and L-b=a (so that the deficiency is a square), that is, a+b=L; in the second case, we are looking for lengths a and b (b>L) such that rect.a,b and b-L=a, that is b-a=L.
126.
It ought to be stressed that, in this case, these are two different solutions: (1) sq.ΔE=rect.AΓ,ΓB = rect.AΓ,(AB − AΓ)=rect.AΓ,AB − sq.AΓ; (2) sq.ΔE=rect.AZ, ZB=rect.\(AZ, (AB - AZ) =\)rect.AZ, AB − sq.AZ.
127.
Let BZ be joined and extended to M on ΓA. Then, since the triangles BZE and MZΓ are congruent (Elem., I.26), BZ = ZM. But since ZΔ is parallel to ΓA we then have immediately (by Elem., VI.2) that AΔ = ΔB.
128.
The word ‘accordingly’ (proinde) suggests that the equality of AΛ,BE and HA, BΘ follows from the equality of AΔ, ΔB. This is true for HA, BΘ (by Elem. III.14); that AΛ is equal to BE, however, follows more directly from the fact that ΛE is perpendicular to ΓΛ and, therefore, parallel to AB.
129.
Elements, III.35: “If in a circle two straight line cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.”
130.
See the note for the 3^rd case.
131.
Elements III.36: “If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut the circle and the other touch it, the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference will be equal to the square on the tangent.”
132.
Since, by Elements III.37 (which is the converse of Elements III.36), if the rectangle ΓΛE is equal to the square on AΔ, AΔ will be tangent to the circle.
133.
Conics VII.15 formulates the proportion alternando: sq.AΓ:sq.ξ::rect.NΓ,MΞ: sq.MN.
134.
A classical argument would be as follows: starting with the proportion, ΓΔ:MΞ::rect.ΓΔ,Φ: sq.MN, Φ is set as a common height to obtain rect.ΓΔ,Φ:rect.MΞ, Φ::rect.ΓΔ,Φ: sq.MN. From this it follows that rect.MΞ,Φ=sq.MN. However, the step in which a common height is taken (so that the ratio is changed from one between lines to one between figures) is generally stated explicitly in classical texts. The immediacy with which Halley states the conclusion leads me to suspect that, as we have seen before, Halley was using an algebraic formulation, that is, he writes ΓΔ ∕ MΞ=ΓΔ ×Φ/MN ², and then “cancels” ΓΔ to obtain 1/MΞ=Φ/MN ² or MΞ ×Φ=MN ².
135.
In other words, the problem of finding MΞ has been reduced to a typical problem of “application of areas,” the subject of the Scholium just prior to proposition XIX.
136.
The application of the 2^nd lemma of the Scholium is clear by comparing the diagrams: E here corresponds to Δ in the Scholium, X to E, M to Γ, Ξ to B. A point corresponding to A is unnecessary since E has been constructed so that EΞ is equal to 1/2(2NΞ+Φ), that is, half the line to which the square on NΞ is to be applied.
137.
That is, (axis + latus rectum) ∕ 2: ξ ∕ 2:: \(\xi /2 : \Phi /2(= {\it { NE}})\).
138.
In other words, whether the arc cuts the axis once or twice inside the hyperbola.
139.
Always written in Greek in the text: thus, .
140.
Conics VII.33 states: “If there is a hyperbola, and the transverse diameter of the figure constructed on the axis is not less than its latus rectum, then the latus rectum of the figure constructed on the axis is less than the latus rectum of [any of] the figures constructed on the other diameters of the section, and the latus rectum of [any of] the figures constructed on diameters closer to the axis is less than the latus rectum of the figures constructed on [diameters] farther from the axis.”
141.
Conics VII.34 demonstrates that the claim of VII.33 holds also when the length of the axis is less than that of its latus rectum, but not less than half of it.
142.
For since latus rectum:axis::AN : AΞ, (latus rectum − axis) : axis : : (AN − AΞ) : AΞ : : NΞ : AΞ. But since latus rectum > 2 ×axis, (latus rectum − axis) > axis > axis:axis, therefore, also NΞ : AΞ > AΞ : AΞ, so that NΞ > AΞ.
143.
Conics VII.35 states (continuing from VII.33 and VII.34): “Furthermore, we make AΓ [the axis (MNF)] less than half the latus rectum of the figure of the section constructed on it: then I say that there are two diameters, [one] on either side of this axis, such that the latus rectum of the figure constructed on each of them is twice that [diameter]; and that [latus rectum] is less than the latus rectum of the figure constructed on any other of the diameters on that side [of the axis]; and the latus rectum of figures constructed on the diameters closer to those two diameters is less than the latus rectum of a figure constructed on a [diameter] farther [from them].”
144.
By Conics VII.6, BK: latus rectum (BK) : : ΞM : MN. But ΞM has been contrived to be equal to NΞ, or \(\frac{1} {2}MN\); therefore, BK must be a diameter which is half its latus rectum.
145.
Recall NE is set equal to Φ ∕ 2 and M is determined by the relation sq.NΞ=rect.\(M\Xi, (\Phi - 2N\Xi )-\)sq.MΞ = rect. \(M\Xi, (2NE - 2N\Xi )-\)sq.MΞ. But NΞ = ΞM, which from the latter relation, implies that NE = 2MΞ. Thus, since \(NE = N\Xi + \Xi E = M\Xi + \Xi E\), we have ΞE = ΞM, that is, E and M coincide.
146.
This of course is a mistake. In the sixth proposition, Halley shows that BK is a mean proportional between MΞ and ΓΔ in the case of an ellipse; it is in the fifth proposition he demonstrates the corresponding fact for the hyperbola.
147.
In other words, for the problem to have a solution, ξ can be as large as one pleases.
148.
Halley’s exposition is not entirely clear. The argument from VII.29 can be formulated as follows. First, observe that by VII.35 there can be unequal diameters having the same latus rectum. Suppose AB and CD are unequal diameters with the same latus rectum L, then, by VII.29, \(\mathrm{rect}.AB,L -\mathrm{sq}.AB = \mathrm{rect}.CD,L -\mathrm{sq}.CD\) or \(\mathrm{rect}.(AB - CD),L = \mathrm{sq}.AB -\mathrm{sq}.CD\), from which it follows that \(L = AB + CD\).
149.
The argument is as in VIII.19.
150.
See the note on the word “analogy” in proposition XI.
151.
The rectangle NMΠ is that contained by NM and MΠ, or MN and \(\mathit{MN} + N\mathit{\Pi }\ (= \mathit{MN} + \Phi \)).
152.
Since \(\mathit{MN} = N\Xi - M\Xi \), we have \(\Xi N + MN + \Phi = \Xi N + N\Xi - M\Xi + \Phi =\)2\(N\Xi + \Phi - M\Xi \).
153.
This is essentially the same as the more conventional separando.
154.
These diorisms follow from Conics VII.21 and 22, respectively.
155.
Again, these follow from the corollaries after Conics VII.31.
156.
Conics VII.16 formulates the proportion alternando, sq.AΓ:sq.(BK − latus rectum(BK)):: rect.NΓ,MΞ:sq.(MN − MΞ). There is no general enunciation for VII.16.
157.
ΞN is given of course because the points Ξ and N are determined by the ratio of the axis to its latus rectum, which is given.
158.
That is, because ΓΔ is given.
159.
No doubt VII.36, and not VII.6, was meant. For Conics VII.36 demonstrates that the difference between the axis and its latus rectum is greater than that between any other diameter and its latus rectum.
160.
By “reciprocally,” Halley means something like “inversely.” Indeed, MΞ:ΓΔ::ΞN:sq.(BK- lr.BK), while ΓΔ and ΞN are fixed. Therefore, as Halley says, MΞ grows continually as BK-lr.BK decreases; hence, while there is an upper bound for BK-lr.BK, namely, the difference between the axis and its latus rectum, there is no lower bound, or, as Halley puts it, the differences “never can reach a minimum.”
161.
By VII.29, \(\mathrm{sq}.BK -\mathrm{rect}.\mathrm{lr}.BK,BK = \mathrm{sq}.\mathrm{Ax} -\mathrm{rect}.\mathrm{lr}.\mathrm{Ax},\mathrm{Ax}\) or \(\mathrm{rect}.BK, (BK -\mathrm{lr}.BK) = \mathrm{rect}.\mathrm{Ax}, (\mathrm{Ax} -\mathrm{lr}.\mathrm{Ax})\). Therefore, if D is the given difference, BK − lr. BK, and D _Ax is the difference between the axis and its latus rectum, Ax − lr. Ax, we have the “reciprocal” relation: BK : Ax : : D _Ax : D. Since Ax, D _Ax, and D are all given, the magnitude of BK is given.
162.
By Conics VII.16, we have Γδ:MΞ::sq.1 ∕ 2(BK-lr.BK):sq.1/2(MΞ − MN). But \(M\Xi = M\Theta + \Theta \Xi = M\Theta + \Theta N\) and \(MN = \Theta N - M\Theta \). Therefore, \(1/2(M\Xi - MN) = 1/2(M\Theta + \Theta N - (\Theta N - M\Theta )) = M\Theta \). So, Γδ:MΞ::sq.1/2(BK-lr.BK):sq.MΘ.
163.
For rect.Γδ,ψ : rect.MΞ,ψ :: sq.1/2(BK-lr.BK) : sq.MΘ. But ψ has been defined so that Γδ:1 ∕ 2(BK-lr.BK)::1 ∕ 2(BK-lr.BK):ψ, that is, so that rect.Γδ,ψ=sq.1 ∕ 2(BK-lr.BK). Thus, sq.1 ∕ 2(BK-lr.BK):rect.MΞ,ψ::rect.Γδ,ψ:rect.MΞ,ψ::sq.1 ∕ 2(BK-lr.BK) : sq.MΘ, from which it follows that rect.Γδ,ψ=sq.MΘ.
164.
Whether it is dividendo or componendo depends on whether M happens to fall between Θ and N or between Θ and Ξ.
165.
The argument is similar to that in Proposition XX: (ΘΞ + ψ)\(-\Theta M = NM + \psi = N\Xi - M\Xi + \psi =\)(NΞ + ψ) − MΞ.
166.
Why Halley puts it this way is unclear: indeed, sq.NΞ and sq.ΞΘ are equally given since NΞ=2ΞΘ or sq.NΞ=4sq.ΞΘ.
167.
Conics VII.37 states: “[In] every ellipse, for the figures of the section constructed on the diameters greater than their [corresponding] latera recta, the difference between the two sides of the figure constructed on the major axis is greater than the difference between the [two] sides of [any of] the figures constructed on the remaining [diameters]; and the difference between the [two] sides of those [figures] constructed on [diameters] closer to the major axis is greater than the difference between the [two] sides of those [figures] constructed on [diameters] farther [from the major axis].
“But in the case when the diameters on which the figures are constructed are less than the [corresponding] lateral recta, the difference between the two sides of the figure constructed on the minor axis is greater than the difference between the [two] sides of the others of these figures; and the difference between the [two] sides of those [figures] constructed on [diameters] farther from it.
“And the difference between the two sides of the figure constructed on the minor [this is Halley’s correction—Toomer notes the error in the Arabic text. (MNF)] axis is greater than the difference between the two sides of the figure constructed on the major axis [again Halley’s correction. (MNF)].”
To understand this proposition—and, therefore, also Halley’s diorisms—it is important to keep in mind that in every ellipse there is a diameter (and of course its equal conjugate) which is equal to its latus rectum. Let us call this diameter E; and so E=lr.E. For all diameters D between E and the major axis, D>lr.D; for all diameters d between E and the minor axis, d<lr.d. Conics VII.37 tells us then that: (1) for those diameters D between the major axis and E, Maj.Ax-lr.Maj.Ax>D-lr.D>D′-lr.D′, where D′ lies farther from the major axis than D; (2) for those diameters d between E and the minor axis, lr.min.ax-min.ax>lr.d-d>lr.d′-d′, where d′ lies farther from the minor axis than d; (3) lr.min.ax-min.ax.>Maj.Ax-lr.Maj.Ax.
168.
This is less transparent than Halley makes it seem, and, as is often the case, I suspect he was working algebraically; I will do the same. Let the given difference be K, and let the two diameters be D and d. Hence, we have the following relations: (1) D-lr.D=lr.d-d=K (the defining relation), (2) D+d=lr.D+lr.d (by subtraction), and (3) D-d+lr.d-lr.D=2K or lr.d-lr.D=2K-(D-d) (by addition). Squaring both sides of the relation (1), we have: D²-2Dlr.D+lr.D²=d²-2dlr.d+lr.d² or D²- d² =2(Dlr.D-dlr.d)+lr.d² – lr.D². But, by Conics VII.30, we have Dlr.D-dlr.d= d²- D², while, by the relations (2) and (3), we have lr.d²-lr.D²=(2K-(D-d))(D+d). Thus, substituting these into the relation D²- d² =2(Dlr.D-dlr.d)+lr.d² – lr.D², we have D²- d²=2(d²- D²)+2K(D+d)-(D²- d²)=2K(D+d)-3(D²- d²) or 4(D²- d²)=2K(D+d), from which the result D-d=1/2K follows immediately.
169.
Again, let the given difference be K, and let the two diameters be D and d. Then since D-d=1/2K, D-1/2K=d. Also, since lr.d-d=K, we have lr.d-d=K=1/2K+D-(D-1/2K)=1/2K+D-d, from which it follows that lr.d=1/2K+D.
170.
Using the same notation as in the last note, since D-d=1/2K, 2D-2d=lr.d-d, or 2D=lr.d+d.
171.
Apollonius states the proposition, which applies both to the hyperbola and ellipse, in the form sq.AΓ:sq.(BK + T)::rect.NΓ,MΞ:sq.(MΞ + MN), where T is the latus rectum for BK. There is no enunciation; Apollonius continues to rely on the figures in Conics VII.6, 7.
172.
It is MΘ which is the excess by which MΞ is greater than ΞΘ.
173.
Where the axis is less than the latus rectum, AΞ is less than ΞΓ, by the definition of the homologue. In this case, then, Ξ lies between M and Θ, so that \(M\Theta = M\Xi + \Xi \Theta \) instead of MΞ − ΞΘ. Therefore, we have the sequence of ratios: MΞ:ΘM::ΘM:ψ; by conversion and permutando, MΞ:ΘM::ΘΞ:ψ − ΘM; again, by conversion, MΞ:ΞΘ::ΘΞ:\(\psi - \Theta M - \Theta \Xi \); finally, substituting ΘΞ + MΞ in place of ΘM and rearranging the terms slightly, we have MΞ:ΞΘ::ΘΞ: (ψ − 2ΘΞ) − MΞ.
174.
Let K be the given sum. Then, as argued at the outset, \(\Gamma \Delta : 1/2K :: 1/2K : \psi \). Therefore, if \(2\Gamma \Delta : 1/2K :: 1/2K\):(third proportional), the third proportional must equal 1/2ψ.
175.
In the first case, where the axis is greater than the latus rectum, the line to which the square on ΞΘ is to be applied is 2\(\Xi \Theta + \psi =\)2ΞΘ + 2ΘE = 2EΞ, while in the second case, where the latus rectum is greater than the axis (and, accordingly, where Ξ lies between E and Θ) the line is ψ − 2ΘΞ = 2ΘE − 2ΘΞ = 2EΞ.
176.
That is, solved.
177.
Conics VII.38 states: “If there is a hyperbola, and the transverse side of the figure constructed on its axis is not less than a third of its latus rectum, then the sum of the lines bounding each of the figures on its diameters [i.e., D+lr.D (MNF)] which are not axes is greater than the sum of the lines bounding the figure constructed on its axis; and the sum of the lines bounding the figures constructed on those [diameters] closer to the axis is less than [the sum of] the sides bounding the figures constructed on those farther from it.”
Apollonius divides the proof of VII.38 into two parts: VII.38 itself considers the case in which Ax ≥ lr.Ax while VII.39 takes up the case in which 1/3lr.Ax ≤ Ax<lr.Ax. Conics VII.40 goes on to show what happens when Ax<1/3lr.Ax.
Specifically, Conics VII.40 states: “If there is a hyperbola, and its transverse axis is less than a third of its latus rectum, then there are two diameters, [one] on either side of its axis, each of which is equal to one third of its [the diameter’s] latus rectum, and the sum of the sides bounding the figure constructed on each of the sides bounding the figure constructed on each of the two is less than [the sum of] the sides bounding any one of the figures constructed on the diameters on that side [of the axis]; and the sum of the sides bounding the figures constructed on diameters closer to [that diameter] is less than [the sum of] the sides bounding the figures constructed on [diameters] farther from it.”
178.
AΞ<1/3ΞΓ or AΞ<1/4AΓ. But AΘ=1/2AΓ. Therefore, ΞΘ=AΘ-AΞ>1/2AΓ-1/4AΓ=1/4AΓ.
179.
It is precisely this that cannot be done where the axis is greater than a third of its latus rectum. But here one has this freedom, and Halley is taking advantage of this to choose ΞE so that a minimum is obtained.
180.
Since \(N\Theta = \Theta \Xi = \Xi E\).
181.
Halley obviously means the 5^th/ proposition.
182.
By definition AΞ : ΞΓ : : Ax:lr. Therefore, ΞΓ + AΞ:ΞΓ::lr+Ax:lr and ΞΓ + AΞ ( = AΓ):2ΞΓ::lr+Ax:2lr, so that AΓ:2\(\Xi \Gamma - (\Xi \Gamma + A\Xi ) :: A\Gamma : \Xi \Gamma - A\Xi :: A\Gamma : N\Xi \)::lr + Ax: lr − Ax. But AΓ:NΞ : : AΘ:ΘΞ. Therefore, AΘ:ΘΞ::lr + Ax:lr − Ax.
183.
Since \(\mathrm{sq}.BK = \mathrm{rect}.A\Theta,\mathrm{lr} -\mathrm{Ax}\) we have \(16 \times \mathrm{sq}.BK = 16 \times \mathrm{rect}.A\Theta,\mathrm{lr} -\mathrm{Ax}\), or, since Ax = 2AΘ, \(\mathrm{sq}.4BK = \mathrm{rect}.8\mathrm{Ax}, (\mathrm{lr} -\mathrm{Ax})\). But the minimum sum, BK + lr_BK has been shown to be precisely 4BK. Therefore, \(\mathrm{sq}.(\mbox{ min. sum}) = \mathrm{rect}.8\mathrm{Ax}, (\mathrm{lr} - Ax) = 8 \times (\mathrm{rect}.\mathrm{Ax},\mathrm{lr} -\mathrm{sq}.\mathrm{Ax}) = 8 \times (\mbox{ figure on}\ \mathrm{Ax}) -\mathrm{sq}.\mathrm{Ax}\). Therefore, \(\mathrm{sq}.(\mathrm{Ax} + \mathrm{lr}) - 8 \times (\mbox{ figure on}\ \mathrm{Ax}) -\mathrm{sq}.\mathrm{Ax} = \mathrm{sq}.(\mathrm{lr} + \mathrm{Ax}) - 8 \times (\mathrm{rect}.\mathrm{Ax},\mathrm{lr} -\mathrm{sq}.\mathrm{Ax} = \mathrm{sq}.\mathrm{lr} - 6 \times \mathrm{rect}.\mathrm{lr},\mathrm{Ax} + 9\mathrm{sq}.\mathrm{Ax} = \mathrm{sq}.\mathrm{lr} - 2 \times \mathrm{rect}.\mathrm{lr}, 3\mathrm{Ax} + \mathrm{sq}.(3\mathrm{Ax}) = \mathrm{sq}.(\mathrm{lr} - 3\mathrm{Ax})\). Hence, \(\mathrm{sq}.(\mathrm{lr} + \mathrm{Ax}) -\mathrm{sq}.(\mathrm{lr} - 3\mathrm{Ax}) = \mathrm{sq}.(\mbox{ min. sum})\).
184.
There will naturally be another diameter on the other side, that is, symmetrically positioned with respect to the axis, which together with its latus rectum will equal the given sum; but there will not be another diameter on the same side of the axis satisfying that condition.
185.
Halley’s formulation is awkward at best. He must mean that, on the one hand, the previous sum (not the given sum), namely, the square of the excess by which the latus rectum of the axis is greater than three times the axis together with the square of the given sum, is greater than the square of the sum of the axis and its latus rectum, while, on the other hand, the given sum itself is less than the sum of the axis and its latus rectum. For only in this case, according to VII.40, will it be that there are four solutions to the problem, two diameters on either side of the axis.
186.
Or BK + BK is one-half the sum of it and its latus rectum.
187.
The argument is similar to that justifying corollary 1 of VIII.25; only one uses VII.29 here, where one used VII.30 there.
188.
Taking D and d to be the diameters, and d+lr.d, we have, by the first corollary, D+d=1/2(D+lr.D) or d=1/2(lr.D-D). As for the second part, we are given d+lr.d=D+lr.D or lr.d=D+lr.D-d. But, by what was just shown, d=1/2(lr.D-D). Therefore, lr.d=D+lr.D-d=D+lr.D-1/2(lr.D-D)=1/2(lr.D+3D).
189.
Conics VII.41 states: “[In] every ellipse, the sum of the [four] sides bounding the figure constructed on its major axis is less than [the sum of] the sides bounding any figure constructed on another of its diameters; and the sum of the sides bounding [one of] the figures constructed on those [diameters] closer to the major axis is less than [the sum of] the sides bounding a figure constructed on [a diameter] farther [from it]; and the sum of the sides bounding the figure constructed on the minor axis is greater than [the sum of] the sides bounding figures constructed on other diameters.”
190.
Suppose D is the axis sought. Then by VII.30, \(\mathrm{rect}.D, (D + \mathrm{lr}.D) = \mathrm{rect}.\mathrm{Ax}, (\mathrm{Ax} + \mathrm{lr}.\mathrm{Ax})\). Therefore, \(D : \mathrm{Ax} :: (\mathrm{Ax} + \mathrm{lr}.\mathrm{Ax}) : (D + \mathrm{lr}.D)\). By concluding that D and D + lr. D are reciprocally proportional, that is, referring to the terms D and D + lr. D only, one wonders whether Halley is actually thinking (here and elsewhere, VIII.23, for example) of a relationship in the form \(D = k/(D + \mathrm{lr}.D)\), a quasi-functional relationship.
191.
As in every analysis, we are supposing the problem done.
192.
See note to proposition IX.
193.
Conics VII.42 states: “The smallest of the figures constructed on diameters of a hyperbola is the one constructed on its axis; and those constructed on [diameters] closer to the axis are smaller than those constructed on [diameters] farther [from it].”
194.
This is proven in Conics I.21. Why Halley points it out, however, is unclear, for it is hardly necessary.
195.
Conics VII.43 states: “The smallest of the figures constructed on diameters of an ellipse is the figure constructed on the major axis; and the greatest of them is the one constructed on the minor axis; and those constructed on [diameters] closer to the major axis are smaller than those constructed on [diameters] farther [from it].”
196.
Conics VII.19 states that sq.AΓ:(sq.BK+sq.T)::rect.NΓ,MΞ:(sq.MN+sq.MΞ), where T is the latus rectum for diameter BK. Again, there is no enunciation.
197.
Applying the sum of squares to the sum of the axis and its latus rectum means having found ψ such that rect.(Ax+lr.Ax),ψ=sq.BK+sq.lr.BK. But (Ax+lr.Ax):MX::(sq.BK+sq.lr. BK):(sq.MN+sq.MΞ). Therefore, taking ψ as a common height, rect.(Ax+lr.Ax),ψ:rect.MX,ψ::(sq.BK+sq.lr.BK):(sq.MN+sq.MΞ). And since rect.(Ax+lr.Ax),ψ=sq.BK+sq.lr.BK, it follows that rect.MΞ,ψ=(sq.MN+sq.MΞ).
198.
Proposition XXIX covers this case and, implicitly, also the third case; proposition XXX covers the second case.
199.
Since rect, MΞ,ψ-2(sq.MΞ-rect.MΞ,NΞ)=sq.NΞ, we have 1/2rect, MΞ,ψ- sq.MΞ+ rect.MΞ,NΞ =1/2sq.NΞ, or rect.MΞ(1 ∕ 2ψ + NΞ)-sq.MΞ=1/2sq.NΞ, which is just the form we need to apply the “application of areas” constructions in the last Scholium.
200.
The shift to 1/2ψ instead of “half ψ” is in Halley’s text.
201.
Indeed, EX = NΠ, while sq.NΠ=sq.ΘΠ+sq.ΘN=2sq.ΘN=1/2sq.NΞ. Furthermore, the radius of the circle \(\Xi E = \Xi \Theta + \Theta E\)=1/2(NΞ + 2ϕ)=1/2(NΞ + 1 ∕ 2ψ). Hence, by the construction given in the 2^nd lemma of the Scholium, 1/2sq.NΞ=rect.MΞ,(2ΞE − MΞ)=rect.MΞ,(NΞ + 1 ∕ 2ψ − MΞ), that is, 1/2sq.NΞ has been applied to the line NΞ + 1 ∕ 2ψ, deficient by a square.
202.
Conics VII.44 states: “If there is a hyperbola, and the transverse side of the figure constructed on its axis is either [1] not less than its latus rectum, or [2] less than it, but [such that] its square is not less than half of the square of the difference between [the transverse side] and [the latus rectum]: then the sum of the squares of the two sides of the figure constructed on the axis is less than [the sum of] the squares of the two sides of any figure constructed on one of its other diameters.” Hence, the diorism is valid both for the case where the axis is greater than its latus rectum—the case treated explicitly here—and also the case where the axis is equal to its latus rectum.
203.
Same reasoning as in the last proposition.
204.
Since MN>MΞ, we may consider the magnitude MN − MΞ, which is NΞ. Thus the word “indeed” (verum) at the start of the sentence. From here, \(\mathrm{sq}.(MN - M\Xi ) = \mathrm{sq}.MN + \mathrm{sq}.M\Xi - 2\mathrm{rect}.MN,M\Xi = \mathrm{sq}.N\Xi \) or \(\mathrm{sq}.MN + \mathrm{sq}.M\Xi = 2\mathrm{rect}.M\Xi,MN + \mathrm{sq}.N\Xi = 2\mathrm{rect}.M\Xi, (M\Xi + N\Xi ) + \mathrm{sq}.N\Xi \).
205.
Since 2rect.MΞ,(MΞ + NΞ)=2sq.MΞ+2rect.NΞ,ΞM.
206.
Having established that rect.MΞ,ψ = sq.MN + sq.MΞ, we have rect.MΞ,ψ − 2rect.MΞ, NΞ − 2sq.MΞ = sq.NΞ, or 1/2 rect.MΞ,ψ − rect.MΞ,NΞ − sq.MΞ = 1/2 sq.NΞ, or again, rect.MΞ, (1/2 \(\psi - N\Xi )-\)sq.MΞ = 1/2 sq.NΞ.
207.
It is understood that as in the previous proposition π has been marked along the minor axis so that \(\Theta \pi = \Theta \Xi = N\Theta \). Therefore, sq.Nπ = 2sq.NΘ = 1/2sq.NΞ.
208.
Conics VII.45 is, in fact, the second case described in VII.44. Conics VII.46, continuing the exposition begun in VII.44, states: “But if the square on transverse diameter [i.e., axis] is less than half of the square of the difference between [the transverse axis] and the latus rectum of the figure constructed on it, then on either side of the axis are two diameters, the square on each of which is equal to half of the square on the difference between [the diameter] and the latus rectum of the figure constructed on it; and the sum of the squares of the two sides of the figure constructed on it is less than [the sum of] the squares of the two sides of any figure constructed on [one of] the diameters drawn on the side [of the axis] on which it lies; and [the sum of] the squares of the two sides of those [figures] constructed on the [diameters] on its side [of the axis] closer to it is less than [the sum of] the squares of the two sides [of figures] constructed on those [diameters] [farther from it].”
209.
Halley is referring to the case of that particular diameter D, for which sq.D+sq.lr.D is a minimum.
210.
Recall 1/2sq.NΞ = rect.MΞ,(1/2\(\psi - N\Xi )-\)sq.MΞ or 1/2sq.NΞ = rect.MΞ,(2\(\Theta E - N\Xi )-\)sq. MΞ = rect.MΞ,(2\(\Theta E - N\Xi - M\Xi \)). Since we are assuming a minimum solution, we must assume that MΞ=2\(\Theta E - N\Xi - M\Xi = \surd \)1/2 \(\times N\Xi = \surd \)2 ×ΞΘ (which is what Halley is referring to by “that which, in square, is twice the square on ΞΘ”). Hence, 2ΘE = NΞ+\(M\Xi + \surd \)2\(\times \Xi \Theta = 2\Xi \Theta + 2\surd 2\times \Xi \Theta \) or \(\Theta E =\Xi \Theta + \surd 2\times \Xi \Theta = (\surd 2 + 1)\Xi \Theta \), that is, ΘE:ΞΘ::(\(\surd \)2+1):1. This, incidentally, also gives further justification for the fact that E, as Halley pointed out, falls “beyond the vertex of the extended axis.”

On another front, it should be pointed out that the root signs appear just as they do above in Halley’s text.
211.
The line ψ is defined so that rect.ψ,(Ax+lr) = (given sum of squares); however, 1/4ψ = ΘE = (\(\surd \)2 + 1)ΘΞ = (\(\surd \)2 = 1)NΞ/2. Therefore, 1/2 (minimum sum of squares) = rect.1/2 ψ,(Ax + lr) = rect.(\(\surd \)2 + 1)NΞ,(Ax + lr).
212.
By definition AΞ:ΞΓ::Ax:lr. Therefore, AΓ( = Ax):NΞ::(ΞΓ + AΞ):(ΞΓ − AΞ)::(lr + Ax): (lr-Ax) or rect.Ax,(lr − Ax) = rect.Ax,lr − sq.Ax = rect.NΞ,(lr + Ax). But it has just been shown that \(2(\surd 2 + 1\))rect.Ax,lr − sq.Ax = 2(\(\surd \)2 + 1)rect.NΞ,(lr + Ax) = min. sum of squares, so, since 2(\(\surd \)2 + 1) = √8 + 2, we have (min. sum of squares):(rect.Ax,lr − sq.Ax)::(√8 + 2):1.
213.
Conics VII.23, stated above in VIII.7, refers to the equality of conjugate diameters; but, as Apollonius himself points out in VII.23, if the conjugate diameters are equal, then (by the definitions following Conics I.16) the latera recta must be equal to the diameters as well.
214.
This of course provides only the magnitude of the diameter; its position can then be found via Proposition V.
215.
The pattern of the argument here runs parallel to that in the first corollary to proposition XXIV and to proposition XXV above. But while those arguments were probably algebraic, as I said, the argument here (and in the first corollary to the next proposition) is undoubtedly algebraic. Let the diameters be D and d, let their latera recta be lr.D and lr.d, and the common sum of the squares of the sides of the figure be K, that is, D² + (lr.D)² = d² + (lr.d)² = K. Call the latter the basic condition. Squaring D² + (lr.D)² and d² + (lr.d)², we obtain (D²)² + 2D²(lr.D)² + ((lr.D)²)² = (d²)² + 2d²(lr.d)² + ((lr.d)²)². Rearranging and factoring, we have
(D² − d² ) (D² + d² ) = 2[d² (lr.d)² − D²(lr.D)²] + [(lr.D)² − (lr.d)²][(lr.D)² + (lr.d)²] (*).
Now, from the basic condition two other relations follow immediately, namely, (lr.d)² − (lr.D)² = D² − d² and (lr.D)² + (lr.d)² = 2K − (D² + d²). Substituting these into (*) and canceling the common factor D² − d² (remember, we are considering two diameters on the same side of the axis for which the sum of squares of the sides of the figures are equal), we have D² + d² = 2K − (D² + d²) − 2[D(lr.D) + d (lr.d)]. From here, we add 2(D² + d²) to both sides and rearrange to obtain 4(D² + d²) = 2K+2[D² − D(lr.D) + d² − d(lr.d)] or D² + d² = 1/2K+1/2 [D² − D(lr.D) + d² − d(lr.d)]. But, by VII.29, D² − D(lr.D) = d² − d(lr.d) = Ax² − Axlr.Ax. Therefore, D² + d² = 1/2K+1/2[2(Ax² − Axlr.Ax)] = 1/2K+Ax² − Axlr.Ax.
216.
In this case, D² + (lr.D)² = Ax² + (lr.Ax)² = K. Therefore, D² + Ax² = 1/2K + Ax²-Axlr.Ax, or D² = 1/2K-Axlr.Ax = 1/2[Ax² + (lr.Ax)²-Axlr.Ax] = 1/2[Ax² + (lr.Ax)²-2Axlr.Ax] = 1/2(Ax-lr.Ax)². As for the second conclusion, from the basic condition, D² + (lr.D)² = Ax² + (lr.Ax)², it follows that (lr.D)² = Ax² + (lr.Ax)²-D². Substituting D² = 1/2[Ax² + (lr.Ax)²-2Axlr.Ax], we have (lr.D)² = Ax² + (lr.Ax)²-1/2[Ax² + (lr.Ax)²-2Axlr.Ax] = 1/2[Ax² + (lr.Ax)² + 2Axlr.Ax] = 1/2(Ax + lr.Ax)².
217.
Since by VII.29, Δ ²-Δlr.Δ = Ax²-Axlr.Ax for any diameter Δ, D²+d² = 1/2K+Ax²- Axlr.Ax = 1/2K+Δ ²-Δlr.Δ. In other words, the diameter Δ can replace the axis in the previous two corollaries.
218.
The construction is as in the previous propositions, namely, a segment ΘΠ, equal to ΞΘ, is cut off the minor axis extended, and ΞΠ is joined: thus, sq.ΞΠ=2sq.ΘΞ=1/2sq.NΞ.
219.
Conics VII.47 and 48 are the analogues for the ellipse of VII.45 and 46. Their statements are as follows:
Conics VII.47: “If there is an ellipse, and the square on the transverse side of the figure constructed on its major axis is not greater than half of the square on the sum of the two sides of the figure constructed on it, then [the sum of] the squares on the two sides of the figure constructed on the major axis is less than [the sum of] the squares on the two sides of [all] other figures constructed on its diameters; and [the sum of] the squares on the two sides of those [figures] constructed on [diameters] closer to it [the major axis] is less than [the sum of] the squares on the two sides [of those figures] constructed on [diameters] farther [from it]; and the greatest of them is [the sum of] the squares on the two sides of the figure constructed on the minor axis.”
Conics VII.48: If there is an ellipse, and the square on its major axis is greater than half of the sum of the [square (MNF—the Arabic text has “squares,” which is wrong of course, and was corrected by Halley in his own edition)] on the two sides of the figure constructed on [the major axis], then there are two diameters, [one] on either side of the axis, such that the square on each of them is equal to half of the square on the sum of the two sides of the figure constructed on it; and [the sum of] the squares on the two sides of the figure constructed on it is less than [the sum of] the squares on the two sides of [any of] the other figures constructed on diameters drawn in that quadrant in which [that diameter] is; and [the sum of] the squares on the two sides of [figures] constructed on those diameters in that quadrant closer to it is less than [the sum of] the squares on the two sides of [figures] constructed on those farther [from it].”
220.
Again, “that equal in square to twice the square on NΘ,” is the line L such that sq.L=2sq.NΘ or L=√2NΘ. So, QE=√2NΘ-NΘ=(√2-1)NΘ.
221.
The square root signs appear in Halley’s text.
222.
For this, and what follows, see the notes for the corresponding section in the previous proposition.
223.
Since by Conics I.15, Ax: ax::ax:lr.Ax (where ax denotes the minor axis, and Ax, as always, denotes the major axis) the figure on the major axis is equal to the square on the minor axis. Therefore, sq.Ax+rect.Ax,lr.Ax=sq.Ax)+sq.ax.
224.
That is, the minimum obtained in the previous paragraph.
225.
Namely, that the sum of squares of the sides of their figures are both equal to the sum of the squares of the sides of the figure of the axis.
226.
By Conics I.15, Ax:ax::ax:lr.Ax and ax:Ax::Ax:lr.ax, from it follows, sq.Ax:sq.ax:: sq.ax: sq.(lr.Ax) and sq.ax:sq.Ax::sq.Ax:sq.(lr.ax). Therefore, sq.Ax:sq.ax::sq.ax:sq.(lr.Ax)::sq.(lr.ax): sq.Ax::[sq.ax+sq.(lr.ax)]:[sq.(lr.Ax)+sq.Ax]. But again, from the proportion Ax:ax::ax:lr.Ax, it follows that sq.Ax:sq.ax::Ax:lr.Ax; therefore, Ax:lr.Ax::[sq.ax+sq.(lr.ax)]:[sq.(lr.Ax)+sq.Ax]. From ax:Ax::Ax:lr.ax, it follows that sq.ax:sq.Ax::ax:lr.ax, so that, in this case, ax:lr.ax:: [sq.Ax+sq.(lr.Ax)]:[sq.(lr.ax)+sq.ax].
227.
The argument is along the same lines as in the first corollary of the last proposition, only here one uses VII.30, where in proposition XXX one used VII.29.
228.
The expressions “same sum of squares” or “same sum” refer to the sum of the squares of the diameter and its latus rectum.
229.
Again, the arguments for all three statements are completely analogous to those which establish the corollaries at the end of proposition XXX.
230.
In order for there to be four diameters having the same sum of squares of the sides of the figure, the sum must be less than that of the squares of the sides of the figure of the major axis, as stated above and in VII.48. But for a diameter D having this sum equal to the sum of the sides of the figure of the major axis, we have from the previous corollary, sq.D=1/2sq.(Ax+lr.Ax), while sq.(lr.D)=1/2sq.(Ax-lr.Ax); therefore, diameter D is still greater than its latus rectum, and, accordingly, must lie closer to the major axis than that diameter which is equal to its latus rectum. The latter, however, is precisely the diameter which is equal to its conjugate: for by Conics I.15, sq.D:sq.D′::D:lr.D for any diameter D and its conjugate D′; therefore, D=lr.D if and only if sq.D=sq.D′, that is, if and only if D=D′.
231.
Corollaries 4, 5, and 6 are not entirely transparent.
232.
Conics VII.20 applies both to the hyperbola and ellipse and states that sq.AΓ:(sq.BK − sq.T)::rect.NΓ,MΞ:(sq.ΞM − sq.MN), where T is the latus rectum for BK (the differences of squares, of course, are reversed where BK < T). As in VII.17 mentioned above, there is no enunciation, and Apollonius continues to rely on the figures in Conics VII.6, 7
233.
For sq.ΞM − sq.MN=rect.(ΞM + MN, ΞM − MN) by Elem.II.6, and ΞM + MN = MN + NΞ + MN = 2MN + 2NΘ = 2MΘ, while ΞM − MN = ΞN = 2ΘΞ.
234.
Conics VII.49 states that, “If there is a hyperbola, and the transverse side of the figure constructed on its axis is greater than its latus rectum, then the difference between the squares on the two sides of that figure is less than the difference between the squares on the two sides of any of the figures constructed on the other diameters; and the difference between the squares on the two sides of those [figures] constructed on [diameters] closer [to the axis] is less than the difference between the squares on the two sides of those constructed on [diameters] farther [from it]; and the difference between the squares on the two sides of any of the figures constructed on diameters which are not axes is greater than the difference between the square on the axis and the figure constructed on it, but less than twice that [difference].

Conics VII.50 states that, “If there is a hyperbola, and the transverse side of the figure constructed on its axis is less than its latus rectum, then the difference between the squares on the two sides of the figure constructed on the axis is greater than the difference between the squares on the two sides of any of the figures constructed on diameters other than it; and the difference between the squares on the two sides of those [figures] constructed on [diameters] closer to the axis is greater than the difference between the squares on the two sides of those constructed on [diameters] farther [from it]; and the difference between the square on any of those diameters and the square on the latus rectum of the figure constructed on it is greater than twice the difference between the square on the axis and the figure constructed on the axis.”
235.
Thus, as before, we are assuming the problem has been solved, and BK is the required diameter.
236.
It is componendo or dividendo, and, accordingly, plus or minus, depending on whether M is between Θ and Ξ or Θ and N.
237.
Conics VII.51 states: The difference between the squares on the two sides of the figure constructed on the major axis of an ellipse is greater than the difference between the squares on the two sides of any figure constructed on other diameters which are greater than the latus rectum of the figures constructed on them [the diameters]; and the difference between the squares on the two sides of those [figures] constructed on those diameters closer to the major axis is greater than the difference between the squares on the two sides of those constructed on those farther [from it]; and the difference between the squares on the two sides of the figure constructed on its minor axis is greater than the difference between the squares on the two sides of any figure constructed on other diameters which are smaller than the latera recta of the figures constructed on them; and the difference between the squares on the two sides of the [figures] constructed on those of these diameters closer to the minor axis is greater than the difference between the squares on the two sides of those constructed on [diameters] farther from it.”
238.
This follows entirely by Conics I.15: Since Ax:ax::ax:lr.Ax and ax:Ax::Ax:lr.ax, we have ax: lr.Ax::lr.ax:Ax. Therefore, sq.ax:sq.(lr.Ax)::sq.(lr.ax):sq.Ax::[sq.(lr.ax)-sq.ax]:[sq.Ax-sq.(lr.Ax)]. But, Ax:lr.Ax::sq.ax:sq.(lr.Ax). Hence, Ax:lr.Ax::[sq.(lr.ax)-sq.ax]:[sq.Ax-sq.(lr.Ax)].
239.
Namely, because Ax>lr.Ax, also [sq.(lr.ax)-sq.ax]>[sq.Ax-sq.(lr.Ax)].
240.
That is, if the given difference of squares is greater than the difference of the squares on the major axis and its latus rectum.
241.
That is, in place of Apollonius’s own analyses and syntheses.

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Program for Science and Technology Education, Ben-Gurion University of the Negev, Marcus Family Campus, Beer-Sheva, 84105, Israel
Michael N. Fried

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Fried, M.N. (2012). Apollonius of Perga’s on Conics: Book Eight Restored or the Book on Determinate Problems Conjectured. In: Edmond Halley’s Reconstruction of the Lost Book of Apollonius’s Conics. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0146-9_7

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