Perspective.

You will say: if 1 mile gives me 15 pairs of hundreds, what will 2 miles give me? It will give me 30 pairs of hundreds, that is hundreds of braccia.

Then in the second way you will say: if every 15 pairs of hundreds gives me 1/15 of the base of the pyramid, which is 48 inches — that base being 4 braccia — what will 30 pairs of hundreds give me? It will give me ...

If the pyramid ... the nature of its base, that is of cold or ... the varieties, because it would freeze.

Reflected motion ... accidents or incorporeal powers ... [may] be much greater than the incident one, that is longer and still more powerful than the simple incident motion, which does not happen in the reflections of corporeal motions.

Whoever had found the ultimate power of the bombard in all its varieties, and had presented such a secret to the Romans, with what speed would they have conquered every land and overcome every army; and what reward could equal so great a benefit? Archimedes, although he had greatly harmed the Romans in the storming of Syracuse, was never denied the offer of the greatest rewards by those same Romans. And in the taking of Syracuse he was diligently sought out, and, found dead, greater lament was made for him by the Roman senate and people than if they had lost their whole army; and they did not fail to honour him with burial and a statue, the leader in which was Marcus Marcellus. And after the second ruin of Syracuse the tomb of Archimedes was rediscovered by Cato among the ruins of a temple, whereupon Cato had the temple and the tomb most honourably rebuilt; of this it is written that Cato said he gloried in nothing so much as in having honoured Archimedes with that adornment.

To see the nature of the planets, open the roof and show the base a single planet; the reflected motion from that base will tell the complexion of the said planet — but see that the base does not see more than one at a time.

Perspective.

The pyramid of one mile: if it is given a base of 4 braccia, and each braccio is divided into 12 inches, so that the 4 braccia give 48 inches, and I divide the length of that pyramid into 15 equal parts in length — the mile being 3000 braccia — then each of the said 15 parts falls to 200 braccia of length. Now I wish to give a rule, from a pyramid of one mile up to a pyramid of 100 miles on the same base of 4 braccia, of how much it narrows at the end of 200 braccia above the base; that is, if the base, as said, was 4 braccia, how much its pyramid will be narrowed at the end of 200 braccia; and so I will continue from the term of 200 braccia of the one-mile pyramid to the term of 2 braccia of the 100-mile pyramid, always taking the width of the pyramid 200 braccia distant from its base.

Upon the base of 4 braccia let 100 pyramids be made, each equally different from the next, the first being one mile, the second two miles, the third three, and so on up to a hundred miles; and let it be given, for each, how much it narrows in diametral thickness at the end of 200 braccia of its length above the base; and what it does in these first 200 braccia it will do in all the other pairs of hundreds; and because the base is set at 4 braccia, and each braccio is 12 inches, they are all 48 inches.

Let it be given how much the pyramid of one mile narrows at the end of 200 braccia distant from the base. Here a rule will be given that serves for all pyramids, the base being set at 4 braccia and each braccio divided into 12 inches, so that the 4 braccia give 48 inches. The mile is made of 3000 braccia, in which those 200 braccia mentioned above go 15 times; therefore, to find that narrowing at the end of 200 braccia above the base of 48 inches, divide that base by the 15th of a mile, that is by the 200 braccia, which yields 1/15 of the whole base — that is, divide those 48 inches by 15, which gives 3 inches and 1/5; and this is how much the pyramid narrows at a distance of 200 braccia beyond the base. And if the pyramid were 2 miles in length beyond the base, which are 30 pairs of hundreds of braccia, divide the 48 inches of the base by that 30, which gives one inch and 3/5; and so continue to the pyramid of 3 miles and 45 pairs of hundreds, and divide the 48 inches of the base by those 45 pairs of hundreds.

And the divisions of the base by 15, the pyramid being one mile, were made in order to make the divisions of the base equal in number to the divisions of the length of the pyramid, because the 15 pairs of hundreds are equal in number to the 15 divisions of that base. Let i e be the base of the pyramid, 4 braccia long, and [let it be] that the divisions of the base are equal in number to the divisions of the length; I say that the part h d, which is in the 4th [quarter] of the pyramid, is narrowed by 1/4 of the base f e, and the part x c narrows another quarter, that is g f, and a b another quarter, and at m the other quarter, i h, is narrowed; and for this reason the base and the axis are divided equally.