(For Aspiring Amateurs)
by Barend Migchelsen

Barend Migchelsen, (pronounced Mikkelsen) learned to sail in The Netherlands in 1943. In 1975 he started to build boats and boat models as a hobby.  Today, he organizes and teaches classroom courses in boat building, and has published several books on the subject.  The following is an excerpt from one of these books.  Click here to check out Barend's books at our store

Often I hear the comment: “Look, it was a long time ago that they tried to teach me mathematics in secondary school.  I never could get excited about it.  I have never needed it later in my life.  Now it looks and sounds like gobblegook to me.  Can’t you do it without any calculations?”  And it may surprise you that the answer is:  Yes, you can if you know what you are doing, and have an eye for the obvious.”

Let me give you an example:

In Algebra, a letter replaces often a very large number.  The purpose of using letters is to replace these big numbers by a few letters and/or symbols and reduce the problem to something that (relatively) anybody can understand.

Here is a little brainteaser:  The letters O, T, T, F, F, S, S, … represent a simple series.  What is the next letter of the series?

If it keeps you awake one week after you read this and still don’t know the answer, you can reach me at 514-631-6431.  But don’t feel bad.  I overlooked the OBVIOUS myself.

Even if you don’t have your own house on a big lot, you still know that an area is determined by its length times its width, or Area = l x w.  You learned that in elementary school.  For a lot of us, it is the only practical application of mathematics that we ever needed.

If a diagonal is drawn in the area rectangle, the area of the two right triangles is a half times length times width, or Right triangle area = ½ x l x w

Fig. 5 - 1

The diagonal is the hypotenuse of a right triangle.  The long leg is the length of the rectangle.  The short leg is the width.

In figure 4-2, the second diagonal crosses the first on exactly at its middle point.  Obvious, a circle with a radius equal to half the length of a diagonal passes through all the four corners.  The diagonals are two middle lines of the circle.  That applies to ANY drawing made that way:

Any two lines drawn from the opposite ends of a middle line of a circle to the same point on the circle line make a right (90º) angle with each other!

Fig. 5 - 2

In the next drawing fig 4-3, the chord is half the actual Overall Length of a double-ender.

The circle arc represents half the actual sheer line arc.  The Overall length is 16 foot, or LOA = 16'.  The constant flare ratio of Profile height to Beam is:  Profile Height/Beam =10/24.  In a double-ender, the half fore of Beam is exactly the same as the half aft of Beam.  Only one half of the sheer line circle arc segment needs to be drawn to get the full picture.  Note the station numbers under the drawing.

Fig. 5 - 3

The actual sheer line in figure 4-4 is exactly the same as In figure 4-3.  On the vertical line h, which is the maximum height of the circle arc, half a circle is drawn.  Any two lines drawn from both ends of this middle line h make a right (90º) angle with each other.

Fig. 5 - 4

Figure 4-5 is a simplified half Body view of the double ender.  At Beam, the half-Breadth is 24"; the Profile height is 10".  The sheer line/hypotenuse is the same line h as in figure 4-4.  The long leg of the right triangle is the half-Breadth at Beam, 24".  The short leg of the triangle is the Profile height at Beam, 10".

Fig. 5 - 5

If this is true for Station #8 (Beam) it is true for all stations

The OBVIOUS conclusion is that on the hypotenuse of the 10"-24"-26" right triangle the heights, h(n) of all the station lines can be set off.  Don’t measure, use a compass, it is much more accurate.  In the right triangle all the measurement of the offset table are shown to two decimal figures accurate.  To keep the drawing clear, only the even stations are shown.  This triangle contains ALL the Profile height and half-Breadth measurements of the sheer line.  Actually, to make the cross frames, it is no longer necessary to make a time consuming half-Breadth view drawing, and/or a Profile view of the sheer line.  These drawings have become redundant.  You probably will agree with me that this simple drawing is easier to read, with less chance of making mistakes than a conventional offset table with its plus (+) and minus (-) signs that make these tables look like ancient hieroglyphs found on tablets from archeological digs in the Middle East.

If the full-sized illustration is made accurately, just take off the measurements with a compass to construct the cross frames.  There is even no need to put the measurements figures in as is done in this illustration.

Fig. 5 - 6

Cross Frames

Besides the measurements for the cross frames, you have to know their exact bevel angles.  That is one of the easiest hurdles to overcome.  The bevel angle of the frames at each station is equal to the center point angle between the radius of the actual sheer line circle arc at station #8 (Beam) and the radius to that particular station.  Angles come pretty accurate, even in a much smaller to scale drawing.  Personally, I always check them with a protractor.  But when it comes to a table saw setting, I make a right-angled template first from any inexpensive stiff material.  How this is done is shown in detail in figure 2-5 in the second article in this series.

Side Panels Fore and Aft

Constant flare angle are easy to cut on a table saw.  To obtain constant flare angles for the stem and the sides of the transom, the side panels are allowed to go straight fore and aft. It is clearly shown in the photograph of the Lunenburg dory on page 8 of the preceding article (#III) in this series.  The side panels continue as the tangent line to the circle arc segment of the sheer line.  And by definition, a tangent line makes a right (90º) angle with the radius at that point of the circle circumference.  When the drawing is made accurately to scale it shows exactly where this tangent line crosses the center line if you need that point for adjustment of the LOA of the hull fore at the bow.  Check figure 3-2 in the preceding article in this series.

Bow and Transom Rake and Length

Every rule has its exception!  Earlier, I mentioned that being able the show the offset table in the form of a simple right triangle drawing as in figure 4–6 make the half-Breadth and Profile drawings redundant.  I still need only two small parts of the Profile drawing to show the rake and the exact length of the bow stem, and the transom board.  But again, this needs only to be drawn as figure 3-5 illustrates.

Last “SNAG

Perhaps you notice that except for the formula AREA = L x W at the beginning of this article not a single other formula is used.  This should come as a relief for all the aspiring amateur boat builders who keep procrastinating to start because of their unfamiliarity with, or adversity to any mathematical subject.

The only reason why you can excuse yourself further is that to be able to start with the drawing, you have to calculate the length of the radius R of the circle segment arc of the actual circle.

Look at the two worked out tables at the end of the third article in this series.  The last column of the first table on page 9 gives you the answer for any type of flare angle you choose for any type of hard-chined boat that can be developed from a 16-foot double-ender.  Once you have made your choice, the figures in the second table provide you with an opportunity to double check the heights of the circle arc segment at each station against your own findings.

Easy Full-Sized Lofting

The successful construction of any (boat-building) project starts with an accurate set of drawings. But how accurate can you make accurate?  Look at figure 5-7.

Fig. 5 - 7

I have a 20-foot measuring tape.  At exactly one inch (1"), I drilled a small 1/8" dia. hole.

I drill, for example, a second hole in the tape at 191.25".  Both holes are not in the middle of the width of the metal tape but only 1/8" from the edge of the strip.  By putting a sharp nail in the first hole and a sharp pencil point in the second hole, I can draw accurately a circle with a 190.25" radius, or any other radius for that matter by drilling an additional hole to place the pencil.

I take an inexpensive 4'x8' panel of Philippine mahogany.  On it, I draw the seven station lines from #1 to #7 inclusive, 12" apart.  The edges of the panel represent the station lines #0 and #8 (Beam).  On this last edge, I mark off 26".  On a straight line on the floor, I line up the tape with the station #8 edge of the panel.  Temporarily, I place the nail in the hole at 191.25" right beside this mark.  On the straight line, I mark the center point of my circle arc segment.  I move the nail to this center point and draw the (half) circle arc segment on the plywood.  If it is done correctly, the line ends at the bottom corner of the other edge, station #0 (= station #16).

When I am satisfied with the accuracy of the actual sheer line arc on the plywood, I draw the radius at each station on the arc but only for a few inches to be able to measure the angle between the station line and the radius for the correct bevel angle setting of the cross frame.

Set of Plans

My full-sized set of plans consists of a panel of plywood with the actual circle arc of the sheer line, and a right triangle with the flare angle between the long (horizontal) leg and the hypotenuse.

Onto the triangle I have only marked and drawn the short legs of the different Profile height of the cross frames that I think are necessary to complete the project.

If the chine line is not parallel to the sheer, I make a Body view drawing.  Last but not least, a preferable full-sized drawing of the Profile view of the rake angle and the length of the bow stem and the transom board.  “That’s all folks!” as they said on the Looney Tunes cartoons.

If any body can show me an easier, faster, and more accurate way, I would like to hear about it.

Sheers and Chines (forever), Barend.

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