CHAPTER VII — Critical Commentary & Notes

On the First Effect of a Line Divided According to Our Proportion

Historical and Philological Commentary

  1. Proposition Citations in Chapter VII:

    • First Effect (Euclid XIII.1): If a line $AB = a+b$ is cut in extreme and mean ratio at $C$ with greater segment $AC = a$, and $AD = \frac{1}{2}AB$ is added to $AC$, then $CD^2 = 5 \cdot AD^2$, i.e., $(a + \frac{a+b}{2})^2 = 5(\frac{a+b}{2})^2$.

    • Campanus VI.19 = Heiberg Euclid VI.30: Pacioli cites la 19. del 6. (“the 19th of the 6th”), which is the numbering of Elements VI.30 (“To cut a given finite straight line in extreme and mean ratio”) in the medieval Latin edition of Campanus of Novara, preceded by Euclid VI.def.3.

    • Euclid II.11: Euclid’s geometric construction in Book II (“To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment”), which constructs the golden section by areas prior to the theory of proportions in Book V.

    • Campanus Euclid IX.16: In Campanus’s commentary on Elements IX.16 (“If two numbers be prime to one another, the second will not be to any other number as the first is to the second”), Campanus proves as a corollary that no three integers can stand in extreme and mean ratio (since if $a+b : a = a : b$ in lowest terms, $a$ and $b$ would be coprime, yet $a+b$ and $a$ would also be coprime, contradicting IX.16).

  2. Christological and Boethian Analogies: Pacioli extends the theological analogies of Chapter V with a Christological comparison (Christ fulfilling the Mosaic Law, Matthew 5:17, and submitting to Mary and Joseph, Luke 2:51) and quotes the famous Neoplatonic hymn O qui perpetua mundum ratione gubernas from Boethius’s Consolation of Philosophy (Book III, metrum 9, line 3): stabilisque manens dat cuncta moveri (“remaining unmoved, He gives all things to move”).

Constantin Winterberg’s Commentary (1889 Edition)

Note 1 (Cap. VII): Equivalence of Pacioli’s Initial Characterization with Euclid’s Definition of Extreme and Mean Ratio (Sachliche Anmerkungen #1, Cap. VII, p. 198, l. 4 from bottom)

Since the definition of continuous division (extreme and mean ratio) according to Euclid (Euclid VI.def.3, Euclid II.11) is stated at the close of Chapter VII—namely, when a given line $ac$ is divided at $b$ into two segments such that the square on the greater segment $ab^2$ equals the rectangle $ac \cdot bc$ contained by the whole line and the lesser segment—we demonstrate its identity with the numerical property stated at the beginning of Chapter VII (Euclid XIII.1):

  • Hypothesis: Let the line $ac$ be divided in extreme and mean ratio at $b$, so that $ab^2 = ac \cdot bc$.

  • Claim: $\left(ac + \frac{bc}{2}\right)^2 = 5\left(\frac{bc}{2}\right)^2$, or equivalently $\left(ab + \frac{ac}{2}\right)^2 = 5\left(\frac{ac}{2}\right)^2$.

  • Proof: By hypothesis, $ab^2 = ac(ac - ab)$, whence

    \[ac^2 = ab^2 + ab \cdot ac \quad \text{and} \quad ab^2 + ab \cdot ac = ac^2.\]

    Adding $\left(\frac{ac}{2}\right)^2$ to both sides yields

    \[\left(ab + \frac{ac}{2}\right)^2 = ab^2 + ab \cdot ac + \left(\frac{ac}{2}\right)^2 = ac^2 + \frac{ac^2}{4} = 5\left(\frac{ac}{2}\right)^2. \quad \text{Q.E.D.}\]

Anmerkung 1 (Cap. VII, S. 198): Identität der zu Anfang gegebenen Definition der stetigen Theilung mit der Euklidischen Definition

Da die Definition der stetigen Theilung nach Euklid gewöhnlich so angegeben wird, wie am Schluss des Capitels gesagt, d. h. wenn die gegebene Linie $ac$ im Punkte $b$ in zwei Abschnitte zerfällt, wovon das Quadrat des grössern $ab^2$ gleich dem Product $ac \cdot bc$ aus der ganzen Länge mal dem kleinern Abschnitte ist, so sei zum Nachweise der Identität der letztern mit der zu Anfang dieses Capitels gegebenen Eigenschaft Folgendes bemerkt:

  • Voraussetzung: Es sei nach der bekannten Euklidischen Definition die Linie $ac$ im Punkte $b$ stetig getheilt, so dass $ab^2 = ac \cdot bc$.

  • Behauptung: $\left(ab + \frac{ac}{2}\right)^2 = 5\left(\frac{ac}{2}\right)^2$.

  • Beweis: Nach Voraussetzung ist $ab^2 = ac(ac - ab)$ oder $ac^2 = ab^2 + ab \cdot ac$. Fügt man rechts und links $\left(\frac{ac}{2}\right)^2$ hinzu, so folgt unmittelbar:

    \[\left(ab + \frac{ac}{2}\right)^2 = 5\left(\frac{ac}{2}\right)^2. \quad \text{w. z. b. w.}\]

Note 2 (Cap. VII): Bisection of the Whole Line and Continuous Division of the Half (Sachliche Anmerkungen #2, Cap. VII, p. 198, l. 14 from bottom)

This corollary (Euclid XIII.1 / Euclid XIII.2) follows immediately if we bisect the line $ab$ at $c$ ($ac = bc = \frac{1}{2}ab$) and divide $bc$ in extreme and mean ratio at $d$; considering the three parts of the line $ab$, the greater segment of the half-line together with the other half satisfies the fivefold square relation $ad^2 = 5\,ac^2$.

Anmerkung 2 (Cap. VII, S. 198): Halbirung der Linie und stetige Theilung der Hälfte

Diese Folgerung ergibt sich, indem man sich die Linie $ab$ in $c$ halbirt, und sodann $bc$ im Punkte $d$ auf die gewöhnliche Art stetig getheilt denkt. Betrachtet man dann die drei Theile, aus denen die Linie $ab$ besteht, so ist der zusammengesetzte Abschnitt $ad = ac + cd$.

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0023, 0024, 0025 (f. 4r–f. 5r)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/007_cap_VII.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/007_cap_VII.txt

  • Marginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.