The Narrowing Lines
The narrowing lines were shown on a plan called the half breadth plan. These lines helped shipwrights draw the individual bends on the body plan. Unlike the later waterlines, they did not show how wide the ship is at various fixed elevations. The elevations of early 17th century narrowing lines varied according to the varying heights of the rising lines. To further complicate matters, the narrowing lines of the floor and the toptimbers did not correspond to a ship’s actual width except at, or very near, the midship bend. The rising and narrowing lines of the floor were also not useful in fairing a ship and, if not well-placed, could even inhibit it. (Lavery, 1984, p. pg 20)
As with the rising lines, separate narrowing lines are drawn fore and aft of the midship bend. The fore and aft ends of the two lower narrowing lines bring the width to zero by definition, and Pett gives us the floor’s width at the midship bend. Hence, we know the widths of the fore and aft ends of both lower narrowing lines. However, one cannot be certain of exactly how the lines narrow. I have based this on the
Treatise. An analogous situation exists for the upper narrowing line where, its aft end is defined by the wing transom's width (which Pett gives us), its fore end has no width, and Pett gives us the ship’s breadth at the midship bend. Again, we do not know exactly how the lines narrow. I have also based this on the
Treatise. The toptimber narrowing line is more speculative. Pett provides no information about it, and it does not have zero width at either end. I have based both the fore and aft lines on the
Treatise, though the fore narrowing lines is more loosely based on it.
The Lower Narrowing Line
The Fore Lower Narrowing Line
The
Treatise says that “the narrowing alow forward on must also be a fuller [line] than the same is aft, that the gripe may be more bluff and stout, which conveys the water with more ease under the bilge and from thence to run without further resistance...”
This narrowing line is another power function; this time to the second power. However, since I narrowed the breadth less than the
Treatise suggests (this is discussed in the later section on the fore upper rising line), I also narrowed the floor less than it suggests. I accomplished this by raising the bend number to the third power.
The Aft lower Narrowing Line
The
Treatise constructs the aft lower narrowing line as an ellipse, explaining that this is because the “narrowing alow aftward on must be by a lank line to the end [that] the water being past [sic] by the broadest part of the bilge at the midship bend may hang but a while there, but may have a passage aftward without cod or cling to dead[en] the way…”.
Calculating the lower narrowing line’s course is more complicated than calculating the course of any other rising or narrowing line. The computational difficulties are compounded because the trigonometry the
Treatise uses is antiquated. Further, some solutions can result in negative numbers, and seventeenth century shipwrights probably did not know what to do with results like this. Even contemporary mathematicians were skeptical, some of whom argued that negative numbers were “unfit in the row of gentlemen [i.e., positive numbers].” (Vyawahare & Agrawal, 2008)
Another problematic aspect of the
Treatise’s calculations is that they cannot be directly applied to any ship whose breadth and keel length are much different from those of the
Treatise’s example ship. In fact, the
Treatise’s calculations are even approximate on its own ship (here, they produce a line that is a little more than 3 ¼ inches too wide). This raises the possibility that if shipwrights actually used this method, the error was allowed to stand, or it had to be corrected by the adze men.
The specific reason the
Treatise’s method is approximate relates to the fact we must extend the line aft of the stern to obtain a line whose width comes to zero at the tuck. The difficulty lies in determining just how far aft we should extend it. I determined this numerically. This is too cumbersome for seventeenth century shipwrights. In fact, as suggested by the approximate solution on the
Treatise’s ship, any exact solution was too cumbersome for them. In fact, finding an “exact” solution with modern mathematical methods is also cumbersome. My line is four ten thousandths of an inch too wide. Since this is unlikely to make much of a difference to modelers (who work at scale), I did not seek a more accurate solution.
References
Lavery, B. (1984).
The Ship of the Line (Vols. Volume II: Design, Construction and Fittings). London: Conway Maritime Press.
Vyawahare, A., & Agrawal, R. K. (2008). A History of Negative Numbers.
Scientia Bruneiana, 9, 75-79.