Dictionary of Fortification

Estimating Unknown Distances Using Similar Triangles Without Instruments

Some nineteenth century engineering manuals and other instructional texts offered a variety of methods that could be used to estimate distances across inaccessible terrain (such as a narrow unfordable river) that could not be accurately measured. Most methods involved setting up a right line distance to be estimated by selecting two points near either edge of the inaccessible terrain and using that line to trace out two similar triangles (that as, triangles that had two angles equal) on ground where they could be measured. When measured the leg of the triangle on accessible ground that corresponded to the leg of the triangle that crossed the inaccessible ground gave a close approximation of the width of the inaccessible ground. Making an estimate using similar triangles was, as it certainly sounds, an exercise in practical geometry that required plenty of level open ground to perform properly since most methods envisioned both triangles being traced to actual size, so at least one leg of the measurable triangle had to be as long as the inaccessible ground was wide. When the inconvenience of restricted open ground was encountered a secondary method could be employed to trace a proportioned triangle that could be measured and the proportion worked out to make the distance estimate.

Four methods found in various period sources for estimating unknown distance across inaccessible terrain have been provided below:

First Method

Select a prominent object (Point A)  such as a large tree or house as the base point on the opposite side of the terrain feature (a river in the example) that prevents an accurate measure from being taken.

Select a second point (Point B) on the near side of the inaccessible terrain feature as the second base point for estimating the unknown distance between the first and second base points.

From Point B advance along a line perpendicular to the direction of the distance between estimated (Line AB) any convenient distance to Point C. Set up a picket at or otherwise mark the position of Point C.

From Point C continue to advance along the same line a distance equal to the distance between points B and C. Mark the terminal point of this distance as Point D.

From Point D advance along a line perpendicular to Line BD until reaching a point (Point E) where the object at Point A is exactly covered by the picket set up at Point C.

Measure the distance between points D and E. By similar triangles (ABC = EDC) the Line DE will nearly equal the unknown distance between points A and B.

Second Method

Select a prominent object (Point A) on the far side of the river or terrain feature that is inaccessible

Select a second point (Point B) on the near side of the inaccessible terrain feature that will serve as the base point for laying out similar triangles that will allow a close estimate of the width of the inaccessible terrain feature (Line AB).
Advance along a continuation of the Right Line AB any convenient distance and mark the stopping point (Point C) with a picket.

From Point C advance along a line that is oblique to Line AC and in the general direction of the inaccessible terrain feature. Mark the stopping point (Point D) with a picket.

Measure the distance between points C and D and mark the halfway point (Point E). In the illustration CE = ED.

Move back to Point B. From Point B advance along a right line toward Point E, carefully measuring the distance between points B and E. On reaching Point E continue along the same line a distance equal to Line BE and mark the stopping point (Point F) with a picket.

Carefully take up a position near Point F at which the picket marking Point F exactly covers the picket at Point D. Advance along the continuation of Line DF  (heading away from the inaccessible terrain) until reaching a point (Point G) at which the picket at Point E exactly covers the object at Point A.

Measure the length of Line FG. By similar triangles Line FG should nearly equal the unknown distance between points A and B.

Third Method

Select an object (Point A) on the far side of the inaccessible terrain to use as a base point for estimating the width of the ground that can not be directly measured.

Select a second point (Point B) on the near side of the inaccessible terrain to use as a second base point and set up a picket at that point. The distance between points A and B taken on a right line will be the distance actually estimated.

Advance along a continuation of Line AB any convenient distance to Point C and mark the point with a picket. Measure the distance between points B and C.

Advance along a right line continuation of Line AC, keeping points A, B, and C carefully aligned, a distance equal to the length of line segment BC (BC = CD). Mark the terminal point (Point D) with a picket.

Advance from Point D along a right line at an oblique angle to the direction of Line AD a distance equal to the length of line segments BC and CD. Mark the distance with a picket (Point E).

From Point E advance along a right line continuation of Line DE a distance equal to the length of Line DE. Mark the end point with a picket (Point F). (BC = CD = DE = EF)

Trace right lines to join the pickets at points B to E and points C to F. This is best accomplished using cord or rope that can be attached to the pickets and drawn tight. Mark the point where the two lines intersect with a picket (Point G).

Move to a position near the picket at Point F. Carefully align the picket at Point F so that it covers the object at Point A. Tie a cord to the picket and have an assistant advance directly toward Point A until he reaches a position close to the edge of the inaccessible ground, making sure that he keeps the cord tight and straight as he advances. Drive a picket at the stopping point and attach the cord to it. It is not necessary to assign any designation to this picket point since the cord only traces the position of Line AF.

Move to a position near Point D and align points D and G. Advance along a right line from Point D to Point G and continue in the same direction until reaching the cord marking Line AF. Set up a picket where these two lines (AF and DG...) intersect (Point H).

Once again using cords to insure greater accuracy mark right lines connecting the pickets at points B to H and E to F and extend those cords along right lines until they intersect. Mark the point (Point K) where they intersect with a picket.

By similar triangles (FHK and BHA) the length of Line FK will closely approximate the distance between points A and B.

Fourth Method

Select an object on the opposite side of the inaccessible ground (Point A).

Select a second point on the near side of the inaccessible terrain (Point B). Line AB approximates the width of the inaccessible ground and will be the distance estimated.

Using a 3-4-5 proportional rope find a line perpendicular to Line AB. Guiding on a distant object advance along the perpendicular line any convenient distance (to Point C), setting up pickets along the way at regular intervals.

Again using the 3-4-5 proportional rope find a line perpendicular to Line BC and advance along that line to a point (Point D) at which any one of the pickets (Point E) on Line BC exactly aligns with the object on the far side of the inaccessible ground (Point A). Measure the lengths of  lines CD, CE, and BE.

By similar triangles triangle EBA will be proportional to triangle ECD. Work out the proportion CE : BE :: CD : AB to estimate the distance between points A and B.

February, 2007