See the first Wikipedia article for the definition of a track transition curve. Typically, tangents are given in degrees not ratios. I think that the ratio given applies to the slope of the transition spiral as it forms a ramp and not to the angle of the tangent (the straight part of the track as it meets the curve).

Tangent 5: a straight line section of a road or railroad (Webster’s)

Wikipedia - Track transition curve, or spiral easement
A Track transition curve, or spiral easement, is a mathematically calculated curve on a section of highway, or railroad track, **where a straight section changes into a curve.** It is designed to reduce the effects of centrifugal force experienced by users. In plan (i.e., the horizontal curve) the start of the transition is at infinite radius and at the end of the transition it has the same radius as the curve itself, thus forming a very broad spiral. At the same time, in the vertical plane, the outside of the curve is gradually raised until the correct degree of bank is reached.
If such easement were not applied the centripetal force needed to change the direction of a rail vehicle would be applied instantly at one point: **the tangent point where the straight track meets the curve**, with undesirable results. With a road vehicle the driver naturally applies the steering alteration in a gradual manner and the curve is designed to permit this, using these principles
http://en.wikipedia.org/wiki/Track_transition_curve

Transition spiral – [too long to post here, but very informative, so I recommend that you read it in its entirety.]
On the early railways it was perfectly adequate to locate the line as a series of tangents joined by flat circular curves. With low speeds, it was unnecessary to superelevate the outer rail to make a comfortable passage of a curve. After a while, traffic would straighten out the first part of the curve a little, sharpening the middle, and this would do quite well.
When speeds increased above about 30 mph, the centrifugal force became uncomfortable to passengers, and the outer rail was subjected to considerable lateral pressure. Superelevation would counteract both of these tendencies, at least approximately. The necessary amount of superelevation could not be introduced suddenly, so a ramp of about 100 ft or so was used, shared between the tangent and the curve. The lateral forces would be even more effective in straightening out the entrance to the curve at these speeds, and an approximate transition curve would result. The location was still done with tangents and circles. Superelevation, incidentally, is a matter of comfort, not safety, since a train is a long way from overturning on any curve that is negotiated without inducing terror.
What is called for is a gradual decrease in radius of curvature R concomitant with the elevation of the outer rail, so that the transition to the circular curve is smooth. **This length of track in which the radius of curvature decreases from infinity to the radius of the circular curve is called the ***transition spiral.*** A spiral in which the superelevation, and therefore the curvature, increases linearly with distance along the spiral has been found to be completely satisfactory.**
http://mysite.du.edu/~jcalvert/railway/transpir.htm

Track curvature (1/ft)
http://www.arema.org/comm/c17/Milwaukee_22.PDF

Long spiral transition lengths also cause difficulty in designing curves with sufficiently long circular arcs in relation to the spiral transitions, especially for smaller deflection angles. Short arc lengths and long transition lengths encourage cutting across the corner into the face of oncoming traffic. Therefore a limit is often placed on the ratio of circular arc length to transition length.
http://www.ipenz.org.nz/ipenztg/conf06/tImages/Tech/11 Weale...

Slope of a road
Main articles: Grade (slope), Grade separation
There are two common ways to describe how steep a road or railroad is. One is by the angle in degrees, and the other is by the slope in a percentage. See also mountain railway. The formulae for converting a slope as a percentage into an angle in degrees and vice versa are: angle =arctan * (slope/100) [taken from graphic]

And slope = 100tan(angle),

where angle is in degrees and the trigonometry functions operate in degrees. For example, a 100% slope is 45°.
**A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10. 1:20, 1:50 or 1:100 (etc.).**
http://en.wikipedia.org/wiki/Slope

--------------------------------------------------
Note added at 2 hrs (2007-11-20 18:15:08 GMT)
--------------------------------------------------

12.2.1 Curve Definition - Prior to discussing curve engineering, one must have a common method of defining a curve. There are two ways of describing curvature in common practice. In North America, a railway curve is described by the angle in degrees subtended by two radii,
whose end points on the curve form a chord of 100 feet in length. **In other parts of the railway world, the length of the radius described above, measured in meters, describes the curve.**…
http://www.arema.org/eseries/scriptcontent/custom/e_arema/Pr...

--------------------------------------------------
Note added at 5 hrs (2007-11-20 21:09:24 GMT)
--------------------------------------------------

Wikipedia - Cant - Rail
In the case of railways, the cant helps a train steer around a curve and keeps the inside wheel edges from touching the rails, minimizing wear.

The amount of cant has to be chosen for a given speed, so that if trains travel through the turn at different speeds, the cant ceases to serve its initial purpose and will actually lead to damage to the wheels going both above and below the original design speed. As a result, a compromise value of cant must be chosen when turns are designed.
The maximum value of cant (the height of the outer rail above the inner rail) for a standard gauge railway is about 6 inches (150 mm).[citation needed]
Ideally, the track should have railroad ties (sleepers) at an increased rate per mile, and a greater depth of ballast to accommodate the increased forces exerted by the moving train.
***When a straight becomes a curve, the amount of cant cannot change from zero to its maximum immediately. Rather, the cant must change (ramp) gradually in a **track transition curve.** The length of the transition curve depends on the maximum speed on the line – the higher the speed, the greater length is required.***
http://en.wikipedia.org/wiki/Superelevation
http://en.wikipedia.org/wiki/Cant_(road/rail)

--------------------------------------------------
Note added at 6 hrs (2007-11-20 21:53:20 GMT)
--------------------------------------------------

Spiral easement See Track transition curve. **Also known as tangent lead-in.**
Track transition curve [this links to a page with the same definition as Wikipedia]: The gradual application of superelevation and tighter curve radius, calculated with reference to the anticipated line speed and the final curve radius, on the approach to a bend. Also known as the transition spiral and spiral easement.
http://www.reference.com/browse/wiki/Rail_terminology


--------------------------------------------------
Note added at 1 day24 mins (2007-11-21 15:59:40 GMT) Post-grading
--------------------------------------------------

You're welcome, Leveleki, and thank you :-)