In the book, Basilica by RA Scotti, architects of the 16th century used a mirror placed on the ground to calculate building height. The book says, "Laying a mirror on the ground so that the top of the building was reflected in its center and measuring the distance from the base of the building to the mirror, he could then calculate the height of the building."

How did he do it exactly? What calculations did he employ?

The measured the distance from the base of the mirror to the building and the angle between the mirror to the ground (it will be the same as the angle between the base and the hypotenuse). Using these 2 you should be able to calculate the height of the building using trig ratios

I don't think you need to use trig here - or you may as well measure the angle of the building directly without the mirror. I believe that this is a simpler technique.

If your mirror is laid flat on the ground, then the ratio of the distance between the building and the mirror and the distance between the mirror and the observer will be the same as the ratio of the height of the building to the height of the observer. This is because the angle of reflection is the same as the angle of incidence, meaning that the triangles are similar (http://en.wikipedia.org/wiki/Similarity_%28geometry%29).

This would give an equation something like:
\frac{d_{building}}{d_{observer}} = \frac{h_{building}}{h_{observer}}

I'll label these variables in a diagram in a minute.

You can measure d building, d observer and h observer easily, giving h building without any trig.

Here, this is what I have in mind, it would work quite well if the distances were measured accurately.

You can see that the triangles are similar, which is what makes the whole thing work. The ground must be flat.