Pantometria — A Geometrical Treatize of Measuring

An ornate decorative initial 'L' featuring floral and vine motifs.

Lift up your quadrant—precisely made—exactly toward the object to be measured. Look diligently through both sights, moving backward or forward as necessary until you see the top, ensuring your line or thin thread falls exactly upon the twelfth point. Now, if you measure the distance from yourself to the base (which I here define as the point directly under the top), then you have the altitude from the highest summit down to the point on the base that is level with your standing position, provided you add the height of your eye to it.

Example: The line and plummet in the figure above falls precisely on the twelfth portion, the distance from you to the base (from A to C) being 15 feet. To this you must add the height of your eye (here imagined as 5 feet), so you have 20 feet, the true altitude from A to B. Just as the length of the cord from the eye to C shows the distance to be measured back, so does the point where the line and plummet touch at C declare where you must begin to measure back.

The 11th Chapter

A woodcut illustration demonstrating a geometrical surveying method. Two figures in historical dress use quadrants to measure the height of a prominent, multi-tiered fortification or castle from two different distances. Lines of sight are drawn from the observers' positions to the top of the tower, labeled with letters A, B, and C to designate the base, summit, and observer locations. The background features a detailed town with gabled houses, a church, and trees.

An ornate decorative initial 'S' containing floral patterns and leaves.

Seek two stations, moving back and forth—either toward or away from the object you intend to measure—so that in one place the thread may fall on the 12th point, and at the other station on the 6th point of the "right shadow." Then, if you double the distance between both places, the summit will appear from that part of the object measured which is equal in height to your eye. Or, if your standing position is level with the base, by adding the height of your eye to that doubled distance, you have the whole altitude from the ground.

If one position causes the thread to fall on 12 and the other on 8 of the "right shadow," then triple the distance to get the height. Or if one falls on 12 and the other on 9 of the "right shadow," then quadruple the distance. Indeed, if one falls on 12 and the other on 6 of the "contrary shadow," then the distance between both stations is equal to the height you are measuring (always understanding this from your eye upward).

Or, if the plummet is forced to fall on 6 points of the contrary shadow, and the other on 4 parts of the same, or in 4 and 3 of the contrary, in all these cases the distance between the places is equal to the altitude. Thus, in measuring the space between the two places, you have obtained the height from your eye upward, adding to it (as I have said) the length from your sight downward, to find the just altitude of the whole, assuming the base is level with your standing.

Example: This figure declares the falling of the thread on 6 of the right, and 12. Also on 12 and 6 of the contrary.