CHAPTER XIV — Critical Commentary & Notes¶
On Its Fifth Marvelous Effect
Historical and Philological Commentary¶
Algebraic Identity Behind the Fifth Effect (Euclid XIII.4): Since $b(a+b) = a^2$ and $b = (a+b) - a$, we have: $$b^2 + (a+b)^2 = b^2 + (a+b)^2 = (a+b)^2 - 2(a+b)b + b^2 + 2(a+b)b = a^2 + 2a^2 = 3a^2$$ In Pacioli's single-radical notation, multiplying $\sqrt{12500}$ by $3$ brings $3^2 = 9$ inside the radical: $\sqrt{9 \times 12500} = \sqrt{112500}$.
Constantin Winterberg’s Commentary (1889 Edition)¶
Note 5 (Cap. XIV): Algebraic Proof of the Fifth Effect (Euclid XIII.4) (Sachliche Anmerkungen #5, Cap. XIV, p. 204, l. 14 from top)
In general algebraic form, for the line $bc$ divided in extreme and mean ratio at $d$ ($cd > bd$):
Hypothesis: $cd^2 = bc \cdot bd$.
Claim: $bd^2 + bc^2 = 3\,cd^2$.
Proof: a) From Note 4, $bc^2 = bc \cdot cd + cd^2$. b) Expanding $bd^2 = (bc - cd)^2 = bc^2 - 2\,bc \cdot cd + cd^2$. Adding (a) and (b) together and substituting $bc^2 - bc \cdot cd = bc(bc - cd) = bc \cdot bd = cd^2$ gives:
\[bd^2 + bc^2 = 2\,cd^2 + bc(bc - cd) = 2\,cd^2 + bc \cdot bd = 3\,cd^2. \quad \text{Q.E.D.}\]
Anmerkung 5 (Cap. XIV, S. 204): Mathematischer Nachweis des fünften Effects (Euklid XIII.4)
In allgemeiner Form unter der früheren Voraussetzung ($cd^2 = bc \cdot bd$):
Behauptung: $bd^2 + bc^2 = 3\,cd^2$.
Beweis: a) $bc^2 = bc \cdot cd + cd^2$ (nach Anm. 4). b) $bd^2 = (bc - cd)^2 = bc^2 - 2\,bc \cdot cd + cd^2$. Durch Addition von a) und b) erhält man:
\[bd^2 + bc^2 = 2\,cd^2 + bc(bc - cd) = 2\,cd^2 + bc \cdot bd = 3\,cd^2. \quad \text{w. z. b. w.}\]
Philological and Folio Apparatus¶
1509 Paganini Witness Folios: 0027 (f. 6r–f. 6r)
Winterberg 1889 Italian Source:
winterberg/01_divina_proportione_italian/014_cap_XIV.txtWinterberg 1889 German Source:
winterberg/03_divina_proportione_german/014_cap_XIV.txtMarginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.