[color=#ff0000] In the hyperbolic plane[/color] given line [i]a [/i]and point [i]P[/i] outside the line. Construct the two lines[br]through [i]P[/i] that are asymptotically parallel to [i]a[/i].  

We follow János Bolyai's [url=http://dide.ker.sch.gr/emekerkyra/books/Bolyai,J%20Theory%20of%20Space.pdf]brilliant construction[/url] (see 34.§, pp. 107-108, 165-166): [br][br][list=1][*]Let [i]T[/i] be the foot of perpendicular from [i]P[/i] to [i]a[/i], and let [i]A[/i] be an arbitrary point of [i]a[/i] that differs from [i]T[/i]. [br][br][/*][*]Construct the quadrilateral [i]PTAF [/i]such that the angles at vertices [i]T, A[/i] and [i]F[/i] are right angles. [br][br][/*][*]Let [i]H[/i] be a point of segment [i]PT [/i]such that [i]AH=FP. [br][br][/i][/*][*]Construct each angle with vertex [i]P [/i]that are congruent to [i]α[/i] = ∠[i]AHT[/i] so that one side of the angle is [i]PT[/i]. The other side of this angle (that differs from [i]PT[/i]) will be asymptotically parallel to [i]a[/i].[br][/*][/list]

The obtained angle has two sides: the perpendicular from P to [i]a[/i], and the line through P that is[br]asymptotically parallel to [i]a[/i]. Bolyai called this angle the [i]angle of snapping[/i] (today the [i]angle of parallelism[/i] is used in English), which is a very suggestive definition. He also found an important relationship between the angle of snapping and the length of the segment [i]PT[/i], which is called [i]distance of parallelism[/i]. [br][br]In mathematics, “showing” some property is usually a synonym of “proving” something. But in our case we will prefer the meaning “visualizing” when doing experiments in the P-model. For instance, by using the means of the P-model it can be shown that the construction is accurate within 10[sup]-8 [/sup]mm. [br][br]We can find further important relationships in this construction. Let [i]A[/i][sub]0[/sub] be the intersection of a circle with center [i]P[/i] and circumpoint [i]F,[/i] and draw the line through [i]P [/i]which is asymptotically parallel to [[i]T[/i],[i]A[/i]). Furthermore let [i]T[/i][sub]0 [/sub]be the perpendicular projection of point [i]A[/i][sub]0 [/sub]on the line ([i]PT[/i]). It can be shown that: [br][br][list][*]rays [[i]T[/i][sub]0[/sub][i]A[/i][sub]0[/sub]) and [[i]PF[/i]) are asymptotically parallel to each other,[/*][*] points [i]A[/i][sub]0 [/sub]and [i]A[/i] are symmetric w.r.t. the perpendicular bisector of segment [i]T[/i][sub]0[/sub][i]T. [/i][br][/*][/list]This relationship [url=https://www.geogebra.org/m/ck6ecca5#material/bmdyhhag]will be used later.[/url]

We constructed the quadrilateral [i]PTAF[/i] with three right angles. It is called a [i]Lambert quadrilateral[/i]. The Swiss mathematician [url=https://en.wikipedia.org/wiki/Johann_Heinrich_Lambert]J. H. Lambert[/url] (1728-1777) did remarkable work on computing the measure of the fourth angle. In Euclidean geometry this angle is also a right angle, while in spherical geometry it is greater than 90°. Lambert considered the option that the fourth angle was less than a right angle, but—unlike Bolyai and Lobachevsky—he eventually rejected it because of the bizarre and contradictory appearance of this quadrilateral from the Euclidean point of view. [br][br]Hopefully the reader holds a different opinion after studying the P-model![br][br][br]