De Sphæra & Cylindro. Lib. I.

These, however, I assume.

I. That the straight line is the shortest of the lines having the same ends.

I delete "surface".

II. If other lines existing in the same plane have the same ends, they are unequal; namely, when both are concave towards the same sides, and either one is entirely contained by the other, and by the straight line having the same ends as it, or some parts are contained, and some are common: and that which is contained is the smaller.

Fig. 4.

As an example, let the lines ACB, ADEB be endowed with these conditions; that is, that they are in the same plane, and have the same ends A, B, and are concave towards the same sides; and ACB is entirely contained by ADEB and the straight line AB, then ACB will be smaller than ADEB. Likewise, the mixed line ZACB is smaller than the line ZADEB, because ZA is common, and the remainder ACB is contained by ADEB, as before.

This statement has hitherto been very poorly received by Editors; they tear it into two, one part lacking truth, the other lacking sense. See Rivaltus, and the others.

III. Similarly, of surfaces having the same ends, if they have their ends in a plane, the one that is flat is smaller.

IV. But if other surfaces also have the same ends, if the ends be in a plane, they are unequal, provided that both are concave towards the same sides, and either one surface is entirely contained by the other, and by the surface having the same ends as it, or some parts are contained, and some are common: and that which is contained is the larger.

Likewise, this Axiom is also very ineptly and absurdly split into two. However, if you have properly perceived the second, you will also easily grasp this one. The examples which will occur more frequently below will shed light on this.

I read "greater" original: "meizon" instead of "between".

Or homogeneous to itself; or of those which can be compared.

Or IV.

V. Furthermore, of unequal lines, unequal surfaces, and unequal solids, the greater exceeds the smaller by that which can (when added to itself) exceed any designated amount having a ratio to itself.

As if the line AC exceeds the line AB by the line BC, the line BC can be taken as many times (or multiplied in such a way) that it exceeds any given line (say ZC). This follows from Def. V of Element V.

These things being supposed,