This section is from the book "The Tinman's Manual And Builder's And Mechanic's Handbook", by Isaac Ridler Butt. Also available from Amazon: The Tinman's Manual And Builder's And Mechanic's Handbook.
This line, as was before observed, is marked L, and its uses are, To Divide a line into any number of equal parts: Take the length of the line by the compasses, and placing one of the points on that number in the line of lines which denotes the number of parts into which the given line is to be divided, open the sector till the other point of the compasses touches the same division on the line of lines marked on the other leg; then, the sector being kept at the same width, the distance from 1 on the line L on the one leg, to 1 on the line L on the other, will give the length of one of the equal divisions of the given line to be divided. Thus, to divide a given line into seven equal parts: - take the length of the given line with the compasses, and setting one point on 7, on the line L of one of the legs, move the other leg out until the other point of the compasses touch 7 on the line L of that leg; this may be called the transverse distance of 7 on the line of lines. Now, keeping the sector at the same opening, the transverse distance of 1 will be the length of one of the 7 equal divisions of the given line; the transverse distance of 2 will be two of these divisions, etc.
It will sometimes happen, that the line to be divided will be too long for the largest opening of the sector; and in this case we take the half, or third, or fourth of the line, as the case may be; then the transverse distance of 1 to 1, will be a half, a third, or a fourth of the required equal part.
To divide a given line into any number of parts that shall have a certain relation or proportion to each other: Take the length of the whole line to be divided, and placing one point of the compasses at that division on the line of lines on one leg of the instrument which expresses the sum of all the parts into which the given line is to be divided, and open the sector till the other point of the compasses is on the corresponding division on the line of lines of the other leg. This is evidently making the sum of the parts into which the given line is to be divided a transverse distance; and when this is done, the proportional parts will be found by taking, with the same opening of the sector, the transverse distances of the parts required. - To divide a given line into three parts, in the proportion of 2, 3, 4: The sum of these is 9; make the given line a transverse distance between 9 and 9 on the two lines of lines; then the transverse distances of the several numbers 2, 3, 4, will give the proportional parts required.
To find a fourth proportional to three given lines: take the lateral distance of the second, and make it the transverse distance of the first, then will the transverse distance of the third be the lateral distance of the fourth; then, let there be given 6:3:: 8, - make the lateral distance of 3 the transverse distance of 6; then will the transverse distance of 8 be the lateral distance of 4, the fourth proportional required.
This sector will be found highly serviceable in drawing plans. For instance, if it is wished to reduce the drawing of a steam engine from a scale of 1 ½ inches to the foot, to another of five-eighths to the foot. Now, in 1 ½ inches there are 12 eighth parts; so that the drawing will be reduced in the proportion of 12 to 5. Take the lateral distance of 6, and keep the compasses at this opening; then open the sector till the points of the composses mark the transverse distance of 12; keep now the sector at this opening, and any measure taken on the drawing, to be copied and laid off on the sector as a lateral distance, - the transverse distance taken from that point will give the corresponding measure to be laid down in the new drawing.
If the length of the side of a triangle, of which we have the drawing, is to be reckoned 45; what are the lengths of the other two sides ? Take the length of the side given, by the compasses, and open the sector till the measure be the transverse distance of 45 to 45; then the lengths of the other sides being applied transversely, will give their numerical lengths.