Department of Mathematics Education

by Jim Wilson, University of Georgia.

See the January, 1997, Mathematics Teacher, for the article on the GLaD construction,
Divide any line segment into a regular partition of any number of parts
by Dan Litchfield and Dave Goldenheim with support from Charles H. Dietrich.

## Part 2

A GSP Sketch

A revised presentation of the GLaD construction results from using the line segment AM as the diagonal of the rectangle, resulting in the following figure and the corresponding simplification of the presentation. This change was mentioned on the GLaD Web Page.
A GSP Sketch

### A Slightly Different Development

A modification of the GLaD construction is suggested by the idea that the interior point leading to P3 is the centroid of a triangle where MB and and AD are extended to meet at X. It is not necessary to keep a right angle at A. The following figure results

Construction

A proof follws readily at each stage considering the similar triangles and the ratio of similarity in each case. All this is very similar to "moving M to C." It presents a single sequence for all cases rather than separate ones for odd and even segmentations.

#### The SaRD construction.

SriRanga Dattatreya presents an alternative construction in which the segment is divided into n parts by means of using easily constructed similar triangles:

### An 1811 Geometry Textbook Presentation

The book, Elements of Geometry, Geometrical Analysis, and Plane Trigonometry, is a textbook written by John Leslie and published in Edinburgh in 1811. There is a note on the title page saying "copious notes and illustrations." This is a textbook and there is no reason to conclude that it contains particularly unique or original material.

On page151 Leslie presents the problem "To cut a given straight line into segments, which shall be proportional to those of a divided straight line." This is the general case of the more usual problem of dividing the line segment into a given number of equal segments. This process is much like the process found in school textbooks.

On pp 151-153, Leslie considers the problem "To cut off the successive parts of a given straight line." He explains this as: " Let AB be a straight line from which it is required to cut off successively the half, the third, the fourth, the fifth, &c." He presents a drawing line the following:

Leslie, of course, accompanies his discussions with proof. The process also leads to a strategy for dividing a line into n equal parts. It is quite like the SaRD construction

Leslie then presents the drawing below

constructed and proved that the segments AF, AH, AK, AM are the half, third, fourth, and fifth parts of the segment AB. This is very similar to the GLaD construction.

On page 424, Note XXXVIII, Leslie comments on the material on pages 150-153 as follows:

The consideration of diverging lines furnishes the simplest and readiest means for transferring the doctrine of proportion to geometrical figures. The order in which Euclid has followed, beginning with parallelograms, and thense passing from surfaces to lines, appears to be less natural.
On page 425, Note XXXIX, Leslie comments
It is will be proper her to notice the several methods adopted in practice for the minute subdivision of lines. The earliest of these -- the diagonal scale -- depending immediately on the proposition in the text is of the most extensive use ..."
The GLaD construction as presented by Dan Litchfield and Dave Goldenheim is a marvelous accomplishment. The fact that it has been around in some form for a very long time should not detract from the rediscovery and modern presentation they have given.