with arbitrary k. This geometrically corresponds to the transformation of one parallelogram to other ones of same area, whose sides touch a hyperbola and both asymptotes. While equation system (1) corresponds to proper Lorentz boosts, equation system (2) produces improper ones. For instance, solving (1) for gives:
Let there be a hyperbola whose asymptotes are AB, BΓ, and let some point Δ be taken in that section, from which ΔΕ, ΔΓ are drawn to ΑΒ, ΒΓ; and let another point H be taken in that section, through which HΘ, HK are drawn parallel to ΔΕ, ΔΖ: I say that the rectangle EΔZ is equal to the rectangle ΘHK.
Let ΔH be joined, and A is connected to Γ. Therefore, since the rectangle AΔΓ is equal to the rectangle AHΓ, it follows that AH is to AΔ as ΔΓ is to ΓΗ. But AH is to ΑΔ as ΗΘ is to ΕΔ, and ΔΓ is to ΓΗ as ΔZ is to ΗΚ; wherefore as ΘΗ is to ΔΕ, so ΔZ is to ΗK: therefore the rectangle EΔZ is equal to the rectangle ΘHK.
If Q, q be any two points on a hyperbola, and parallel straight lines QH, qh be drawn to meet one asymptote at any angle, and QK, qk (also parallel to one another) meet the other asymptote at any angle, then HQ·QK = hq·qk. Let Qq meet the asymptotes in R,r. We have RQ.Qr=Rq.qr; therefore RQ:Rq=qr:Qr. But RQ:Rq=HQ:hq, and qr:Qr=qk:QK; therefore HQ:hq=qk:QK, or HQ.QK=hq.qk.
In the next proposition XIII, Apollonius showed that if a line is drawn parallel to the asymptotes, within the space between asymptotes and hyperbola, it must meet the hyperbola exactly once. In his demonstration, Apollonius used the previous proposition XII when comparing the area of several parallelograms whose sides are drawn parallel to the asymptotes.[M 3][M 4]
The ratios given by Apollonius:
represent an equation system that can be solved for ΔΕ, ΔΖ, resulting in the squeeze mapping:
producing in line with Apollonius result that rect. EΔZ is equal to rect. ΘHK.
In case ΔΕ, ΔΖ, HK, ΘΗ are all drawn parallel to the respective asymptotes, it follows u'=ΔΕ, v'=ΔΖ, u=HK, v=ΘΗ, k= and therefore Apollonius result becomes equivalent to Lorentz boost (9a), signifying squeezed parallelograms located between the asymptotes and the hyperbola.
In general, the identity demonstrates the invariance of the area of all parallelograms that are constructed in line with the proposition XII, thereby representing all points of a hyperbola defined by HK·ΘΗ = const. That is, the invariant area HK·ΘΗ = const. together with const=1 gives HK=1/ΘΗ, which implies that ΘΗ is inverse proportional to HK. Thus when HK is increased into k·HK using some factor k, it follows that ΘΗ must be proportionally diminished into ΘΗ/k in order to preserve invariance of area.
It can be seen that Mercator's relations c and d implicitly correspond to hyperbolic functions cosh and sinh (which were explicitly introduced by E:Riccati (1757) much later). In particular, his result AH·FH=AI·BI and AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or in modern notation, corresponds to squeeze mapping or Lorentz boost (9a) as well as (9e) in terms of η because:
The case of squeezing a given square or parallelogram as a means to generate hyperbolas was discussed by w:Euclid Speidell (1688):[M 6]
[..] from a Square and an infinite company of Oblongs on a Superficies, each Equal to that Square, how a Curve is begotten which shall have the same properties and affections of an Hyperbola inscribed within a Right Angled Cone
[..] There is a Square ABCD, whose Side or Root is 10, let DB be prolonged in infinitum, and continually divided equally by the Root, or DB, and those Equal Divisions numbered by 10, 20, 30, 40, 50, 60, 70, &c. in infinitum: Upon these Numbers let Perpendiculars be erected, which call Ordinates, and each of those Perpendiculars of that length, that Perpendiculars let fall from the aforesaid Perpendiculars to the Side or Base CD (which call Complement Ordinates) the Oblongs made of the Ordinate Perpendiculars, and Complement Ordinate Perpendiculars may be ever Equal to the Square AD, which is easily done thus, for it is &c. produces the Length of the Ordinate Perpendiculars
[..] all the Oblongs made of the Ordinates, and Complement Ordinates are each of them equal to the Square AD, which is here 100
[..] the like Demonstration serves for all the Oblongs or Parallelograms standing upon the Base CD, by the Tips or Angular Points of those Parallelograms, or from the Ends of all the Ordinates standing upon 20, 30, 40, 50, 60, 70, in infinitum, draw the Curve Line from A towards E, so shall you describe the Curve AEFGS [..].
This corresponds to squeeze mappings (9a) with u=v=10 and k=1,2,3,4,5,6,7,..., thus u'v'=uv=100.
But it is to be acknowledg'd, that many Properties of an Hyperbola are better known from another manner of generating the Figure; which Way is this: Let LL and MM be infinite Right Lines intersecting each other in any Angle whatever in the Point C: From any Point whatever, as D or e, let Dc, Dd, be drawn parallel to the first Lines, or (ec, ed), which with the Lines first drawn make the Parallelograms as DcCd, or ecCd; Now conceive two sides of the Parallelogram as Dc, Dd, or ec, ed, to be so mov'd this way and that way, that they always keep the same Parallelism, and that at the same time the Area's always remain equal: That is to say, that Dc and ec remain always Parallel to MM, and Dd or ed always Parallel to LL; and that the Area of every Parallelogram be equal to every other, one Side being increas'd in the same Proportion wherein the other is diminish'd. By this means the Point D or e will describe a Curve-Line within the Angle comprehended by the first Lines;
"The system of equations determines all points of the curve , because and being given numbers, each arbitrary value of gives a point of this curve. The elimination of the indeterminate between equations (2) will therefore lead to the equation of the curve in question. This curve is therefore a hyperbola related to its asymptotes ."
This is equivalent to (improper) Lorentz transformation (9a-2).
w:Charles-Ange Laisant extended circular trigonometry to elliptic trigonometry. In his model, polar coordinates x, y of circular trigonometry are related to polar coordinates x', y' of elliptic trigonometry by the relation[M 10]
He noticed the geometrical implication that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.
This is equivalent to Lorentz transformation (9a).
w:Sophus Lie (1879/80) derived an operation from w:Pierre Ossian Bonnet's (1867) investigations on surfaces of constant curvatures, by which pseudospherical surfaces can be transformed into each other.[M 11] Lie gave explicit formulas for this operation in two papers published in 1881: If are asymptotic coordinates of two principal tangent curves and their respective angle, and is a solution of the Sine-Gordon equation , then the following operation (now called Lie transform) is also a solution from which infinitely many new surfaces of same curvature can be derived:[M 12]
In (1880/81) he wrote these relations as follows:[M 13]
In (1883/84) he showed that the combination of Lie transform O with Bianchi transform I produces w:Bäcklund transformB of pseudospherical surfaces:[M 14]
As shown by Bianchi (1886) and Darboux (1891/94), the Lie transform is equivalent to Lorentz transformations (9a) and (9b) in terms of light-cone coordinates 2s=u+v and 2σ=u-v. In general, it can be shown that the Sine-Gordon equation is Lorentz invariant.
w:Edwin Bailey Elliott (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:[M 27]
The second line is equivalent to squeeze mapping or Lorentz boost (9a) as well as (9e) in terms of η.
Equations (1) together with transformation (2) gives Lorentz boost (9a) in terms of light-cone coordinates. Transformation (3) is equivalent to Lorentz boost (9b) in terms of Bondi's k factor. Eisenhart's angle σ corresponds to ϑ in (9e).
↑Apollonius/Halley (1710), Prop. XII of book II on p. 114; Latin: "Si ab aliquo puncto eorum, qua sunt in sectione, ad asymptotos duæ rectæ lineæ in quibuslibet angulis ducantur, & ab alio quovis puncto in sectione sumpto ducantur aliæ rectæ his ipsis parallelæ : rectangulum sub parallelis contentum æquale erit contento sub rectis ipsis quibus ductæ fuerant parallelae.
Sit hyperbola, cujus asymptoti AB, BΓ sumatur in sectione aliquod punctum Δ, atque ab eo ad ΑΒ, ΒΓ, ducantur ΔΕ, ΔΓ; sumatur autem & alterum punctum H in sectione, per quod ducantur HΘ, HK ipsis ΔΕ, ΔΖ parallelæ: dico rectangulum EΔZ rectangulo ΘHK æquale esse.
Jungatur enim ΔH, & ad A, Γ producatur. itaque quoniam rectangulum AΔΓ aequatur rectangulo AHΓ; erit ut AH ad AΔ ita ΔΓ ad ΓΗ. sed ut AH ad ΑΔ ita ΗΘ ad ΕΔ, & ut ΔΓ ad ΓΗ ita ΔZ ad ΗΚ; quare ut ΘΗ ad ΔΕ ita ΔZ ad ΗK: rectangulum igitur ΕΔZ rectangulo ΘHK est æquale."
↑Apollonius/Heath (1896), Proposition 34 (Apollonius, Book II, Prop. 12).
↑Apollonius/Halley (1710), Prop. XIII of book II on p. 114-115
↑Apollonius/Heath (1896), Proposition 35 (Apollonius, Book II, Prop. 13).
↑Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).