The Works of Euclid, Vol. 1

I never gave a "good to print" without having first ensured that all corrections had been made. By means of an errata, which I will place at the end of the last volume, one will be able to correct the errors that a very attentive reading of the printed work, which I shall conduct, will have led me to discover.

M. Nicolopoulo, of Smyrna, a man commendable for his rare talents and a very skillful proofreader, was kind enough to read a great number of my proofs. M. Patris, who has long cultivated the Greek, Latin, and French languages, took infinite pains to ensure that my edition would do honor to the French presses; while reading the proofs, he took care to compare the Latin and French versions carefully with the Greek text, and to make marginal observations for me.

Among the variants of this first volume, there are some that deserve to be noted in particular.

In all the Greek and Latin editions, the 4th, 5th, and 6th postulates are placed among the common notions.

The demonstration of the 7th proposition of Book I has two cases, and yet only one case is demonstrated in all the manuscripts without exception, and in the editions of Basel and Oxford. The second case is that in which point Δ falls within the triangle ΑΒΓ, or alternatively point Γ within the triangle ΑΒΔ. The demonstration of the second case requires it to be demonstrated beforehand that, if the equal sides of an isosceles triangle are extended, the angles below the base are equal to one another; and this is what Euclid did in proposition 5, and which he only did for the sake of proposition 7, since, apart from that, this demonstration is no longer necessary in the rest of the Elements of Euclid; from which it follows evidently, say all the commentators, that the Greek text of the demonstration of proposition 7 is truncated. All the commentators were wrong. The figure was incomplete in all the manuscripts and in all the editions. I drew a second figure; I extended the straight lines ΒΓ, ΒΔ, and the demonstration was found to be complete, without my having changed a single word of the Greek text.

The demonstration of the 24th proposition of Book III has three cases. Indeed, point Α being on point Γ, and point Β on point Δ, it must be demonstrated...