Pantometria — A Geometrical Treatize of Measuring

When the thread falls on 12 portions, the shadow is equal to the object itself. In 6 of "right shadow," it is but half; on the side of "contrary shadow," it is twice the height. To conclude, you may see that as the side in "right shadow" exceeds the parts, so does the altitude of the body exceed the shadow, and the contrary occurs in "contrary shadow." Behold your figure: the thread cuts 6 parts of contrary shadow in the quadrant next to the right hand. The shadow B C being 210 feet, multiply (as I have said) the length of the shadow, 210 feet, by 6 (the parts cut by the thread), which increases to 1260. Divided by 12, it results in 105—the altitude of such a body that had a shadow of 210 feet. Thus it is for all such cases.

The 9th Chapter.

Of Vegetius concerning heights

A woodcut illustration demonstrating the measurement of heights using shadows. It features a fortified tower on the left with the number "105" next to it, and a landscape of hills on the right. Two suns are depicted: one in the upper left casting rays onto the tower and ground, and another on the far right. Figures of people are seen in the valley, and the number "210" is marked on the ground level.

Decorative initial letter 'H' in a woodcut frame, featuring two classical figures, one seated and one standing with a club.

He affirms that by any certain measure of a thing standing directly (whose shadow is known), the height of any other thing (whose shadow is then measured) may be found by the rule of proportion.

Suppose the shadow of any thing is 210 feet. Now say: if 20 (the shadow of a thing known) gives a height of 10, what shall 210 give? It results in 105, the height.

The 10th Chapter.

Without shadow or any calculation, to take approachable heights by your geometrical quadrant.

A large woodcut diagram illustrating how to use a quadrant to measure the height of a building without shadows. On the left is a tall, ornate building with architectural details and statues, labeled point 'B' at the top and 'A' at the base. On the right, a man kneels (point 'C') and sights through a large quadrant towards the top of the building. A sun with a face looks down from the upper right. Points on the diagram are labeled A, B, C, N, and O, and a numerical scale is visible on the quadrant.