UNIT II — MATHEMATICAL & ALGEBRAIC SUMMARY

Exact Symbolic Digest of Cossic Powers, Surds & the Six Quadratic Canons

1. Cossic Power Hierarchy (fols. 143r–149v)

Syncopated 1494 Cossic Lexicon

1494 Print	Vernacular / Latin Term	Algebraic Power	Geometric Model
.n°.	Numero (dragma)	$x^0 = 1$	Known area / constant magnitude
.co.	Cosa (res, radix)	$x^1$	Side of the unknown square
.ce.	Censo (census)	$x^2$	Unknown square area
.cu.	Cubo (cubus)	$x^3$	Solid cube on the cosa
.ce.ce.	Censo de censo	$x^4$	Biquadratic power $(x^2)^2$

2. Pacioli’s Six Canonical Equations (fols. 145r–146r)

Implemented symbolically in geometor.pacioli.proportions.solve_cossic_quadratic() for positive coefficients $a, b, c > 0$ (normalized with $p = b/a$, $q = c/a$):

Six Positive-Coefficient Cossic Canons

Canon	1494 Cossic Form	Positive Root Formula	Euclidean Basis
1	$a.\text{ce.} = b.\text{co.}$	$x = b / a$	[I.cn.3]
2	$a.\text{ce.} = c$	$x = \sqrt{c / a}$	[II.14]
3	$b.\text{co.} = c$	$x = c / b$	[VII.19]
4	$a.\text{ce.} \; \bar{p} \; b.\text{co.} = c$	$x = \sqrt{(p/2)^2 + q} - p/2$	[II.4] (Gnomon)
5	$a.\text{ce.} \; \bar{p} \; c = b.\text{co.}$	$x = p/2 \mp \sqrt{(p/2)^2 - q}$ (if $(p/2)^2 \ge q$)	[II.5]
6	$b.\text{co.} \; \bar{p} \; c = a.\text{ce.}$	$x = \sqrt{(p/2)^2 + q} + p/2$	[II.6]

3. Equidistant Gradation Criterion (fols. 149v–150r)

• Reducible Higher Trinomials: Any equation $a x^{2m} + b x^m = c$ (e.g., $m=2$, censo de censo e censo equale a numero) reduces via $y = x^m$ to Canon 4, 5, or 6, yielding $x = \sqrt[m]{y}$.

• Irreducible Cubics (`fol. 150r`): Non-equidistant forms $x^3 + px = q$, $x^3 = px + q$, and $x^3 + q = px^2$ lack quadratic reduction; Pacioli marks them unsolved in general as of November 1494.