Notes on the Elements

We elucidate briefly on the philosophical and conceptual content contained in Book 1 of Euclid’s elements. We present a selection of the Propositions therein, with the diagrams reproduced, and the proofs re-worked in modern language.

Subtle points, such as the philosophical category of angle, are looked at in relation to Euclid’s conceptualization. In this case, the three-fold division of this category (quanta, qualia, relation), is primarily conceptualized from a relational perspective. The concept of degree, for example, is not defined, but the notion of right angle (quantity) is necessary.

Of primary importance is the realization of the basic form of mathematical structures. In this case, a priori notions are existent and defined at the outset, and constructive structures are built from this vantage point. A circle exists, and is defined. Its definition is not equivalent to its existence. However, Proposition 1, for example, defines an equilateral triangle, and proves its existence. There is no such proof for that of a circle. Thus philosophical existence precedes proof and definition.

Note the use of the parallel postulate in proving Proposition 32. Further, refer back to Proposition 16, which proves that the exterior angle is greater than either of the interior and opposite angles, along with the relational comparison of angles utilized in the proof:

Archimedes’ Proof of the Circular Area

We consider the proof of Archimedes – exhausting the circle from above and below by a sequence of polygon approximations; this construction is compared to that of a most basic integral on the real line.

In the second note we consider notions of analysis including Cantor’s countability, i.e. equivalence at infinity, along with a brief note on the epsilon-analysis of Weierstrass. We then move into a development of the fundamental theorem of calculus.


The style is cursive script 行書. The proverb reads, 德高望重, and refers to a notion of moral vision.

The piece has been dated according to the traditional calendar: 庚子小寒 – here. Written by Jonathan Root – 風乾 (note that 乾 is written in the style of 歐陽詢, cf. 九成宮醴泉銘碑).