This little instrument was exhibited in a somewhat crude state at the meeting of the British Association at Newcastle in 1889. It has since been modified in several respects, and improvements suggested by practical use have been introduced, bringing it into a practical form, and enabling a much greater accuracy to be attained. The principle is one which is occasionally employed for setting out circles with a pocket sextant, viz., the property of a circle that the angle in a segment is constant. The leading feature of the invention is the arrangement of scales, which enables the operation of setting put large curves for railway or other work to be carried out without requiring any calculations, thereby enabling any intelligent man to execute work which would otherwise call for a knowledge of the use of a theodolite and the tables of tangential angles.

The Trotter Curve Ranger 832 05 fig1

The instrument is intended to be thoroughly portable; so much so, indeed, that it is not necessary or even desirable to use a tripod. It may be held in the hand like a sextant, or may be carried on a light staff. The general appearance is shown in Fig. 1. It will be seen that a metal plate, on which two scales are engraved, carries a mirror at one end and an eye piece at the other. The mirror is mounted on a metal plate, which is shaped to a peculiar curve. A clamp and slow motion provide for rapid and for fine adjustment. The eye piece is set at an angle, and contains a half silvered mirror, the upper portion being transparent. This allows direct vision along the axis of the eye piece, and at the same time vision in another direction, after two reflections, one in the eye piece and the other at the adjustable mirror. Fig. 2 is an outline plan of the instrument when closed. In the first form of the instrument only one mirror was provided, but by the double reflection in the improved pattern, any accidental twisting of the rod or handle produces no displacement of the images, since the inclination of one mirror neutralizes the equal and opposite inclination of the other.

No cross line is required with the new arrangement, since it is only necessary that the two images should coincide.

The Trotter Curve Ranger 832 05 fig2

The dotted line A B represents the direct ray, and the line A C D the reflected one. Fig. 3 shows the different geometrical and trigonometrical elements of the curve, which can be read upon the various scales, or to which the instrument may be set. An observer standing at C sights the point B directly and the point A by reflection. A staff being set up at each point, he will see them simultaneously, and in coincidence if the instrument be properly set for the curve. If any intermediate position be taken up on the curve, both A and B will be seen in coincidence. If the two rods do not appear superimposed, the operator must move to the right or the left until this is the case. The instrument will then be over a point in the curve. Any number of points at any regular or irregular distances along the curve can thus be set out. One of the simplest elements which can be taken as a datum is the ratio of the length of the chord to the radius, AB/AO, Fig. 3. This being given, the value of the ratio is found on the straight scale on the body of the instrument, and the curved plate is moved until the beveled edge cuts the scale at the desired point.

The figure of this curve is a polar curve, whose equation is r = a ± b sin. 2 θ, where a is the distance from the zero graduation to the axis of the mirror, and b is the length of the scale from zero to 2, and θ is the inclination of the mirror. In the perspective view, Fig. 1, the curved edge cuts the scale at 1. The instrument being thus set, the following elements may be read either directly on the scales or by simple arithmetical calculation:

The Trotter Curve Ranger 832 05 fig3
FIG. 3

The radius = 1.

AB, the chord, read direct on the straight scale.

AFB, the length of the arc, read direct on the back or under surface of the plate.

FH, the versed sine, read direct on the curved scale.

ACB, the angle in the segment, read direct on the graduated edge.

EAB, the angle between the chord and the tangent, read direct on the graduated edge.

GAB, the tangential angle = 180 deg. - ACB.

AOB, the angle at the center = 2GAB.

AGB, the angle between the tangents = 180 deg. - AOB.

OAB, the angle between the chord and the radius = EAB - 90 deg.

GF = - - - FH.

The foregoing elements are contained in a very simple diagram, Fig. 4, which is engraved on the instrument, together with the following references:

B = 180 deg. - A.

C = 2B.

D = 180 deg. - C.

E = A - 90.

Only one adjustment is necessary, and this is provided by means of the screws which fix the inclination of the eyepiece. This is set at such an angle that the instrument, when closed and reading 90° on the divided limb, acts as an optical square.

It is not necessary, as in the ordinary method with a theodolite, that one end of the curve should be visible from the other. If an obstacle intervenes, all that part of the curve which commands a view of both ends can be set out, and a ranging rod can be set up at any point of the curve so found, and the instrument may be reset to complete the curve.

To set out a tangent to the curve at A, Fig. 3, set up a rod at A and another at any point C, and take up a position on the curve at some point between them. Adjust the mirror until the rods are seen superimposed. Then moving back to A, observe C direct, and set up a rod at E in the line observed by reflection. Then A E is the tangent required. Similarly, on completing the setting out of a curve, and arriving at the end of the chord, the remote end being seen by reflection, the direction observed along the axis of the eyepiece is the new tangent.

Any of the angles or other ratios already mentioned may be used for setting the instrument, but if no data whatever are given, as in the rough surveys for colonial railways where no previous surveys exist, it is only necessary to select points through which the curve must pass, to set up ranging rods either at the extremities of the desired curve, or at any points thereon, to take up a position on the desired curve between two rods, and to adjust the instrument until they are seen in coincidence. The curve can then be set out, and fully marked, and the elements of the curve can be read on the scales and recorded for reference.

The Trotter Curve Ranger 832 06 fig4

Various other cases which may occur in practice can be rapidly met by one or other of the various scales. Suppose the angle A G B between the tangents be given, together with the middle point F on the curve, Fig. 3. Subtract this angle from 180 deg., the difference gives the angle at the center A O B. Take half this, and set the instrument to the angle thus found. Walk along the tangent until a rod set up at some point in the tangent, say E, is seen in coincidence with a rod set up at B. The position of the instrument then marks the point of departure A. A rod being placed at A, the first half of the curve may be set out; or, if B is invisible, the instrument may be reset for the angle E A B, and the whole curve set out up to B. No cutting of hedges is necessary, as with theodolite work, for a curve can easily be taken piece by piece. Inclination of the whole instrument introduces no appreciable error. If the eye piece be pointed up or down hill, the instrument is thrown a little to one side or other of the tip of the staff, but in a plane tangent to the circle. Errors made in setting out a curve with the Trotter curve ranger are not cumulative, as in the method of tangential angles with a theodolite.

No corrections for inaccurate hitting of the final rod can occur, for the curve must necessarily end at that point. It should be observed that the instrument is not intended to supersede a theodolite, but it has the great advantage over the older instrument that no assistant or chains or trigonometrical tables or any knowledge of mathematics are required. The data being given, by a theodolite or otherwise, an intelligent platelayer can easily set out the curve, while the trained engineer proceeds in advance with the theodolite. No time is lost; as in chaining, since the marks may be made wherever and as often as convenient. In work where high accuracy is required this instrument is well adapted for filling in, and where a rough idea of the nature of a given curve is required, the mirror being adjusted for any three points upon it, the various elements may be read off on the scales. A telescope is provided, but the errors not being cumulative, it is rarely required. The curve ranger weighs 1 lb. 10 oz., and is manufactured by Messrs. Elliott Bros., St. Martin's Lane, London. It is the invention of Mr. Alex. P. Trotter, Westminster. - The Engineer.