Elements: Book II Propositions 12 and 13 (II.12 and II.13)
Euclid's Scissors: Discussion

Ching-Kuang Shene (冼鏡光), Professor Emeritus
Department of Computer Science
Michigan Technological University
Houghton, MI 49931
USA
Created April 5, 2026

We discussed Euclid's Scissors on a previous page.

You may wonder why Euclid did not use the scissors invented by him to prove Propositions II.12 and II.13. I guess no one can provide a definitive answer to this question. If you read Elements carefully, you may find an interesting fact that only one proof is given to each proposition. Just like any author writing a textbook, usually s/he develops some theory and proves some results from that theory in a chapter and most likely s/he uses the obtained results to prove more results. If this assumption is true, we have a possible explanation. In Book II of Elements, Euclid developed the theory of areas and naturally he used the theory of areas to prove II.12 and II.13. Also note that papyrus scroll, the media used by Euclid, was perhaps expensive. As a result, Euclid did not want to write all books in which each proposition would receive multiple proofs.

According to Sir Thomas L. Heath, on page 404 of The Thirteen Books of Elements, Vol. 1: Books 1-2, Grégoire de Saint-Vincent in 1646 published a book Opus Geometricum Quadraturae Circuli et Sectionum Coni Decem Libris Comprehensum of more than 1,000 page. PP. 31-32 as shown below proved II.12 and II.13 using the Euclid's Scissors technique.

Page 31
Page 32

Heath argued that the original proofs of II.12 and II.13 are proofs; however, Euclid's Scissors is more like an algorithm (or procedure). Euclid's Scissors provided a step-by-step construction for proving II.12 and II.13. Additionally, proofs and algorithms are equally important.

In summary, Euclid developed an algorithm using Euclid's Scissors, but did not use it to prove II.12 and II.13. When an angle becomes \( 90^{\circ} \), propositions II.12 and II.13 combined reduced to I.47, the Pythagorean Theorem. Textbooks in the late 19th century started to present the Euclid's Scissors and some even included both proofs. For example, you may want to take a look at the following textbooks:

  1. Julius Henrici and Peter Truetlein's 1881 book Lehrbuch der Elementar-Geometrie (p. 97 and pp. 99-100).
  2. Charles Smith and Sophie Bryant's 1901 book Euclid's Elements of Geometry: Books I-IV, Vi and XI (pp.142--145).
  3. H. M. Taylor's 1893 book Euclid's Elements of Geometry: Books I-IV, VI and XI (pp. 158--161).
Given these facts, Euclid could be the first person who formulated and proved the Law of Cosines.

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