UNIT I — PROLOGUE TO MASTER DOMINICUS, PISAN METROLOGY & DISTINCTIO I–II¶
Epistle to Master Dominicus, Euclidean Principles, Agrarian Units of Pisa, Rectangular Fields, and Square Roots
Source Text: fibonacci:pg-prol, fibonacci:pg-1, fibonacci:pg-2 (Boncompagni Vol. II, pp. 1–30 · fol. 1r–18v) · Latin (1220 / Boncompagni 1862) · Summary · GEOMETOR Notes
Epistle & Prologue to Master Dominicus¶
Latin Locus: Prologus (p. 1) · Commentary: Notes on the Prologue
You requested, my friend Dominicus and reverend Master, that I compose for you a book on the practice of geometry, containing both geometrical demonstrations and the practical usages of measurement according to the custom of surveyors. Moved therefore by the affection of your friendship, and wishing that this work be profitable equally to those who desire to work according to geometrical demonstration and to those who proceed according to lay custom (secundum vulgarem consuetudinem), I have labored to set forth this treatise clearly and in order, for which reason I have divided it into eight distinctions (in octo distinctionibus ipsum divisi):
First Distinction (fibonacci:pg-1): On the measurement of equilateral and rectangular quadrangular fields, with the foundational rules of Pisan land measure and the propositions of Euclid’s second and third books needful to this art.
Second Distinction (fibonacci:pg-2): On the finding of square roots (de inventione radicum) of integers, fractions, and metrological subdivisions.
Third Distinction (fibonacci:pg-3): On the measurement of all planar fields (in mensuratione omnium camporum)—triangles, quadrilaterals, multi-angular polygons, and circles.
Fourth Distinction (fibonacci:pg-4): On the division of fields among co-heirs or partners (consortes).
Fifth Distinction (fibonacci:pg-5): On the finding of cube roots (de radicibus cubicis inveniendis) and their binomial operations.
Sixth Distinction (fibonacci:pg-6): On the dimension of corporeal solids (in dimensione corporum)—columns, pyramids, cones, spheres, and the five regular solids inscribed in a sphere.
Seventh Distinction (fibonacci:pg-7): On finding the heights of elevated things, depths, and planar distances through the quadrant (quadrans).
Eighth Distinction (fibonacci:pg-8): On certain geometrical subtleties (de quibusdam subtilitatibus geometricis) pertaining to the pentagon, decagon, and irrational lines.
Introductory Definitions & Common Conceptions¶
Latin Locus: Introductoria (pp. 2–3) · Commentary: Notes on Definitions & Pisan Metrology
Before entering upon the first distinction, certain principles, petitions, and common conceptions of the mind (communes animi conceptiones) from the first book of Euclid (euclid:elements) must be set down beforehand:
Point (Punctus): A point is that which has no part. [I. Def. 1]
Line (Linea): Indeed a line is length without breadth, whose extremities are two points; and a straight line is that which lies evenly between its points, namely the shortest extension from one point to another. [I. Def. 2–4]
Surface (Superficies): A surface is that which has breadth and length only, whose boundaries are lines; and a plane surface is that which lies evenly between straight lines. [I. Def. 5–7]
Rectilinear Angle (Angulus planus): A plane angle is the mutual inclination of two lines touching one another in a plane and not lying in a straight continuation. When a straight line standing upon a straight line makes the adjacent angles equal to one another, each of the equal angles is a right angle (angulus rectus), and the standing straight line is called the perpendicular or cathetus upon that on which it stands. [I. Def. 8–10]
Solid Body (Corpus): A solid is that which possesses length, breadth, and depth (or height), whose boundaries are surfaces. [XI. Def. 1]
The Agrarian Metrology of Pisa (De Mensuris Pisanis)¶
Latin Locus: De Pertica Pisana (pp. 3–4) · Commentary: Notes on the 6 × 6 Pertica Lattice
Now the Pisan rod (pertica), wherewith lands are measured in our city, consists of six linear feet (pedes), and each linear foot contains eighteen ounces or points (unciae sive puncta $.XVIII.$); thus one linear pertica contains $6 \times 18 = 108$ linear ounces.
When, however, a square field ABDC has one linear pertica on every side—so that AB and CD each measure $6$ linear pedes—surveyors do not call $1 \text{ pes} \times 1 \text{ pes}$ a superficial foot, lest the fractional parts depart from the duodecimal custom of the rod. Rather, they divide the square ABDC by strip-measure:
Superficial Pes (pes): A rectangular strip ABP2P1 of one linearpertica in length ($6$ linear pedes) by one linear pes in breadth, bounded below by the dividing line P1P2. Six such strips fill the whole superficial pertica ABDC.
Superficial Uncia (uncia): A strip of $1$ linear pertica by $1$ linear uncia, so $18$ superficial unciae make $1$ superficial pes, and $6 \times 18 = 108$ superficial unciae make $1$ superficial pertica.
Superficial Denarius (denarius): The unit square CU1U3U2 of $1$ linear pes by $1$ linear pes at corner C, bounded by the grid lines through U1 and U2. Thus $6$ denarii compose one superficial pes strip ABP2P1, and $6 \times 6 = 36$ denarii compose the entire superficial pertica ABDC, whose perpendicular median axes EF and GH meet at the center I and bisect the square equally.
At the estate scale, five and one-half superficial perticae ($5\frac{1}{2}$ perticae) make one panorum (a basket-measure of seed-land), and twelve panora make one Pisan stariorum, which is therefore forty-six and twenty—that is, sixty-six superficial perticae ($12 \times 5\frac{1}{2} = 66$ perticae, or $2{,}376$ denarii).
Distinctio I — Rectangular Fields & Euclidean Geometric Algebra¶
Latin Locus: Distinctio I (pp. 5–18) · Commentary: Notes on Distinctio I
If a square .a.b.c.d. has $2$ perticae on every side, draw through the midpoint of each side parallel lines dividing the square; four unit squares of $1$ pertica each are produced ($2 \times 2 = 4$). Likewise, if an oblong rectangle .a.b.g.d. has breadth $2$ perticae and length $3$ perticae, dividing the sides into unit perticae produces $2 \times 3 = 6$ superficial perticae.
When lengths are given in mixed perticae, pedes, and unciae, cross-multiplication obeys the positional hexagesimal-duodecimal table:
Perticae into perticae yield superficial perticae;
Perticae into pedes yield superficial pedes ($\frac{1}{6}$ pertica);
Pedes into pedes yield denarii ($\frac{1}{36}$ pertica, or $\frac{1}{6}$ superficial pes).
If a farmer wishes to lay out a rectangular plot of exactly one Pisan stariorum ($66$ superficial perticae) along a frontage of $7$ perticae, divide $66$ by $7$: the required depth is $9\frac{3}{7}$ perticae, which in surveyor’s subunits is $9$ perticae, $2$ pedes, and $10\frac{2}{7}$ unciae, since $7 \times \left(9 + \frac{2}{6} + \frac{10\frac{2}{7}}{108}\right) = 66$.
To ground all subsequent mensuration and quadratic equations upon unshakable proof, Leonardo appends the rectangular identities of Euclid’s Book II (euclid:elements) and the chord theorem of Book III:
If a straight line be cut at random into two parts $a$ and $b$, the square on the whole is equal to the squares on the parts and twice the rectangle contained by the parts: $(a + b)^2 = a^2 + b^2 + 2ab$. [II. 4]
If a straight line .ab. be cut into equal parts at .g. and into unequal parts at .d., the rectangle contained by the unequal segments .ad. and .db., together with the square on the intermediate segment .gd., equals the square on the half .gb.: $ad \cdot db + gd^2 = gb^2$. [II. 5]
If .ab. ($10$) be bisected at .g. ($5$) and another line .bd. ($3$) be added to it in a straight line, then $ad \cdot db + bg^2 = gd^2$ ($13 \times 3 + 25 = 64 = 8^2$), and the sum of the squares $ad^2 + db^2 = 2(ag^2 + gd^2)$ ($169 + 9 = 2(25 + 64) = 178$). [II. 6, 10]
Distinctio II — On the Finding of Square Roots (De Inventione Radicum)¶
Latin Locus: Distinctio II (pp. 18–30) · Commentary: Notes on Square-Root Extraction
Because diagonals of rectangles, altitudes of triangles, and sides of regular figures frequently fall upon numbers that are non-square (numeri non quadrati), the surveyor must know how to extract square roots both in Hindu-Arabic positional digits (as taught more fully in Liber Abaci) and in Pisan subdivisions.
Exact Positional Algorithm: Drawing upon $(10a + b)^2 = 100a^2 + 20ab + b^2$ [II. 4], group the digits in pairs from the right. Subtract the greatest square $a^2$ from the highest pair, double the root found ($20a$) as divisor for the remainder, determine the next digit $b$ such that $(20a + b)b$ does not exceed the remainder, and continue through all pairs.
First & Second Rational Approximations for Surds: When a remainder $r = N - a^2 > 0$ remains after the integer part $a$, the root lies strictly between $a + \frac{r}{2a + 1}$ and $a + \frac{r}{2a}$. Taking $a_1 = a + \frac{r}{2a}$, its square exceeds $N$ by $\left(\frac{r}{2a}\right)^2$; subtracting $\frac{(r / 2a)^2}{2 a_1}$ yields a second rational convergent accurate to small fractions of an uncia:
$$a_2 = a_1 - \frac{a_1^2 - N}{2 a_1}.$$
Roots of Agrarian Fractions: To find the side of a square plot whose area is $\frac{2}{3}$ of a superficial pertica, reduce $\frac{2}{3}$ pertica to unit squares: $\frac{2}{3} \times 36 = 24$ denarii (square pedes), or $24 \times 324 = 7{,}776$ square unciae. Since $88^2 = 7{,}744$ and $88$ linear unciae equal $4$ pedes and $16$ unciae (with remainder $32$, giving $\frac{32}{176} = \frac{2}{11}$ uncia), the side of the square plot is $4$ pedes and $16\frac{2}{11}$ unciae.