The Works of Euclid, Vol. 3

But allow me to note that Euclid could not have arrived at his goal by the same path as Archimedes, had he not utilized the four principles or postulates which are present at the beginning of the book On the Sphere and Cylinder; yet Euclid had not admitted these four postulates. Wherefore Euclid, who demonstrated that circles are to one another as the squares of their diameters, did not demonstrate that the circumferences of circles are to one another as their diameters, and that a circle is equal to a triangle whose base is equal to the circumference and whose altitude is equal to the radius; for it would have been necessary for that reason that Euclid had admitted, just as Archimedes did, that the sum of two tangents drawn from the same point is greater than the arc comprehended by them, etc.

Proposition LXXXVI of the Data, which is LXXXVII in my edition, had not a little perplexed the most learned Mr. Gregory. In his preface, he states that this theorem is very greatly corrupted, and that he could not restore it with the aid of the manuscripts. I believe that his error arose from the fact that he was unaware of the lemma which follows Proposition LXXXVI of my edition, and which I shall here set forth in a not dissimilar manner.

Two geometric diagrams illustrating proportions within parallelograms and rectangles. The left diagram shows a rectangle with vertices and points labeled E, A, Δ, Γ, Z, H, and Θ. The right diagram shows a similar geometric construction with points Δ, E, A, H, B, Θ, Z, and Γ. Lines and segments demonstrate perpendicularity and length relationships discussed in the text.

Let AΓ be a parallelogram; through point B let a straight line EZ be drawn perpendicular to BΓ; let ΔA be extended; let BZ be made equal to BA; let the rectangles ΓE and ΓZ be completed, and from any point H on AB let HΘ be drawn perpendicular to BΓ. Therefore, as parallelogram ΓA, that is, rectangle ΓE, is to rectangle ΓZ, so will BE be to BZ. But as BE is to EZ, that is, as BE is to BA, so is the sine HΘ of the angle ABΓ to the radius BH; therefore, as parallelogram ΓA is to rectangle ΓZ :: sine ABΓ : R.

From this it is manifest that whatever the lengths of the sides AB, BΓ of the parallelogram AΓ may be, the rectangle ZΓ will be given in magnitude as long as the angle ABΓ remains the same, and as long as the parallelogram AΓ does not cease to be equal to the given surface.