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 once (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 be freezing.

The reflected motion ... accidents or incorporeal powers ... to be much greater than the incident, that is longer and even more powerful than the simple incident motion, which does not happen in the reflection of corporeal motions.

Whoever had discovered the ultimate power of the bombard in all its varieties, and presented such a secret to the Romans, with what swiftness would they have conquered every land and overcome every army; and what reward could equal so great a benefit? Archimedes, although he had greatly damaged the Romans at the storming of Syracuse, was never denied the offer of the greatest rewards by those same Romans. And at the taking of Syracuse he was diligently sought out, and, found dead, greater lamentation was made in the Roman senate and people than if they had lost their whole army; nor did they fail to honour him with a burial and a statue, of which Marcus Marcellus was the head. And after the second ruin of Syracuse the tomb of that Archimedes was found by Cato among the ruins of a temple, whereupon Cato had the temple and the tomb rebuilt most honourably; of which it is written that Cato said he gloried in nothing so much as in having honoured that Archimedes with that ornament.

To see the nature of the planets, open the roof and show a single planet to the base; the motion reflected from that base will tell the complexion of the said planet, but see that such a base does not view 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 once (so the 4 braccia give 48 once), and I divide the length of that pyramid into 15 equal parts of length — the mile being 3000 braccia, to each of the said 15 parts falls 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: 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 is narrowed at the end of 200 braccia; and so I shall continue from the term of 200 braccia of the pyramid of one mile to the term of 2 braccia of the pyramid of 100 miles, always taking the width of the pyramid 200 braccia distant from its base.

On the base of 4 braccia let 100 pyramids be made, each differing equally from the next: let the first be 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, and its 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 once, they are all 48 once.

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 once, so that the 4 braccia give 48 once. The mile is made of 3000 braccia, into which the said 200 braccia enter 15 times; therefore, to find that narrowing at the end of 200 braccia above the base of 48 once, divide that base by the fifteenth of a mile, that is by the 200 braccia, which gives 1/15 of the whole base — that is, you will divide those 48 once by 15, which gives 3 once 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 is 30 pairs of hundreds of braccia, divide the 48 once of the base by that 30, which gives one once and 3/5; and so continue to the pyramid of 3 miles and 45 pairs of hundreds, dividing the 48 once 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 pyramid's length, 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 — the divisions of the base being equal in number to the divisions of the length — I say that the part h d, which is in the 4th part of the pyramid, is narrowed by 1/4 of the base f e; the part x c narrows another quarter, that is g f; 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.