The sector and the related proportional compasses (proportional dividers) were developed in the second half of the 16th century. There are several inventors, most of them from Italy but most people attribute its development to Galileo Galilei. The sector consists of two arms connected by a pivot joint. For example, linear, trigonometric, and logarithmic scales were engraved to the two bars. Computations work by solving problems with similar triangles (ie all the angles are the same therefore the sides must be proportional).
The sector was a very useful instrument at a time when artisans and military men were poorly educated in mathematics and, often, were unable to perform even elementary arithmetical operations. Much like modern business people use computer spreadsheets to do computations they may or may not fully understand, artillerymen and surveyors of the 18th Century used the sector to do basic arithmetic quickly and mechanically. The inaccuracy of the analog scales on the sector were trivial when compared to the utility. Determining that 3∙7≈20 (approximation) was just as useful as determining 3∙7=21 (exact solution).
Suppose you want to divide a line six inches long into five equal parts. Measure off six inches with dividers, and open the sector until the distance between the 5 and 5 on the linear scales is six inches apart. By the principle of similar triangles, the distance between 1 and 1 on the linear scales is then one fifth of the distance of the distance between five and five. Measure the distance 1 to 1 with dividers and the five equal lengths can be measured off the original six-inch line.
Of course, this is very elementary arithmetic and most surveyors would do this in their heads but what if your problem was more complex. This is where all the other lines on the sector come to play.
The Geometric Lines: these help us find the side of a plane figure that has a given ratio to another similar polygon. For example, given the triangle ABC, we wish to find the side of another triangle that has the [area] ratio to it of 3:2. Select two numbers in the given ratio; let these be for instance 12 and 8. Taking line BC with a compass and opening the Instrument, fit this to points 8-8 of the Geometric Lines; then, without changing the opening, take the distance between points 12-12. If we now make a line of that length the side of a triangle, corresponding to line BC, the surface will doubtless be three-halves that of triangle ABC.” The lengths on the Geometric Lines vary as the square root of the labeled values. If L represents the length at 50, then the generating function is: f(n) = L(n/50)1/2, where n is a positive integer less than or equal to 50.
The Stereometric Lines: Given one side of any solid body we can use these to find the side of another that has a given volume. For example, let line A be the diameter of a sphere, or to speak more familiarly a ball, or the side of a cube or other solid, and let it be required to find the diameter or side of another similar solid having to it the ratio which 20 has to 36. Take line A with the compass, and opening the Instrument fit that to points 36-36 of the Stereometric Lines; this done, take next the distance between points 20-20, which will give line B, the diameter (or side) of the solid which is to the other, whose side is A, in the given ratio of 20 to 36.” If L is the scale length at 148, then the scale generating function is: f(n) = L(n/148)1/3, where n is a positive integer less than or equal to 148.
The Metallic Lines: These Lines have divisions to which are affixed these symbols: Au, Pb, Ag, Cu, Fe, Sn, Mar, Sto, which mean Gold, Lead, Silver, Copper, Iron, Tin, Marble, and Stone. From these you can get the ratios and differences of weight found between the materials thus designated. The intervals between any correspondingly marked pair of points will give the diameters of balls or sides of other solid bodies similar to one another and equal in weight in another material.
This simple tool, soon became the “pocket calculator” of field engineers in the 17th, 18th and 19th Centuries.
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