...contrary, by doubling the space between the two first places, the altitude appears. In like manner, the distance from the middle station to the last brings the height from your sight.

How vertical lengths are known.

I would not have you ignorant here of how to determine lengths that are in height, which are not easy to reach. First, by your art mentioned before, get either height; subtract the lesser height from the greater, and by force your desired length remains. Or thus: let the plummet fall under 12 portions, mark your place, and go in toward the object (the thread remaining as it was) until you see the base of that length; the distance between the two standing positions is undoubtedly the length. No example is needed here.

The 12th Chapter

How to quickly know all accessible heights by your quadrant with calculation.

A large decorative initial 'Y' featuring ornate foliate designs and a small human or creature face mask at the top center.

Your quadrant, as said before, is handsomely elevated against or toward the object to be measured, perceiving through the sights nothing more than the top. Mark well the division of points touched on your scale. If they are of the "right shadow," multiply the distance from you to the base by 12 and divide by the parts of your scale that your thread made manifest. But if they are of the "contrary shadow," work conversely: that is, multiply by the parts and divide by 12, remembering always to add the height of your eye downward to your quotient. So have you your desire, the base being equal with your standing.

Example.

Admit the thread with the plummet notes 6 parts of the contrary, as you may see in this figure: the distance from the base A to your standing B is 115 feet. Multiply that by 6, so you have 690, which, divided by 12, yields 57 1/2 feet. To this, add 5 feet (being the height of your sight to the ground) to conclude the altitude is 62 1/2 feet.

Here you shall note that in the measurement of heights with this instrument, it is requisite first to find what part of the altitude is level with your eye, which you may do thus:

Cause the plummet and thread to fall upon the side line of your quadrant where the degrees begin. Then, searching through the sights, that part which you can see of your altitude (the plummet hanging upon the aforementioned lateral line) is level with your eye. The height of this point from the base, compared with the height of your eye, reveals the inequality or difference of the ground—that is to say, how much higher or lower the base of the object to be measured is than the ground at your station. This difference, as you shall see cause to add or subtract from your heights found as previously declared, yields the most exact true altitude. Thus you shall be assured never to err, no matter how uneven or unlevel the ground may be.