Euclids Circle can be divided by the ordinal plane and the abscissa plane. The euclidean circle is divided into 360° and four quadrants of 90°. Pythagorean trigonometry is essential in carpentry and is based upon the 90° right triangle.

There are 360° in the latitudinal plane of a sphere (like earth), and also 360° in the longitudinal plane of the earth. The 360° of navigation is very similar to the 360° circle of Euclid and Pythagoras, and therefore very similar to carpentry.

Armillary Sphere

From the origin of a Euclidean circle, the ordinal and abscissa vectors of a quadrant proceed to the two fix locations at the perimeter of the circle. Pictured above is the triangle formed within a mathematical premise circle with a 3 inch radius. The entire premise circle is being divided by the ordinal and abscissa vector values. For carpentry, the 45°, 45°, 90°, triangle is formed basically by putting any two like numbers (pictured: 3”) on the framing Square against a piece of wood to be cut, and it is the most fundamental Carpentry and Carpentry Square triangle. It has 45° and 90° angles. A framing square, therefore, easily lays out a square 90° and also 45° cuts. No also, that roof pitch of 12/12 is a 45° peak angle because it rises 12” for each 12” of run. Rise and run, and total rise and run, can be important in stairs including landscaping considerations as well as in roof pitch.

The square rule pictured is a 90° angle. It is easy to read the 3” and the 4” marks on the square rule. The “invisible“ 5 inch distance is shown using an architects scale. Lines where a board in the shop or on site is being marked essentially mimic the angles and the linear proportion of the 3, 4, 5 triangle or the 45° 45° 90° triangle, so angles and lines are always proportional to the trigonometric form; when the square rule is placed on a piece to be drawn, measured and cut. This is true to scale too, so a framing square can even lay out angles across entire sheets of plywood, or even if a truss is being laid out upon a space the size of an entire shop floor or in a parking lot. Trigonometry allows all of the distances and angles to be double checked during layout.

Speed squares offer easier ways of working with angles that are acute or obtuse relative to the basic trigonometry of the 45°, 45°, 90°, and the 3, 4, 5, triangular forms of Pythagorean trigonometry. Speed squares remain capable of working with right triangles, and, SOH, CAH, TOA, and Asquared plus B squared equals C squared Pythagorean theorem triangles.

A note on crown mouldings in millwork.

First of all, beveled crown mouldings are not 45° angle cuts in their three dimensional state. The 45°, 45°, 90° jig that I made after many failures truly helped me learn how to make beveled millwork cuts. At first, the level, plumb and moulding angles were confusing to me when perfecting my mitres. Yet, as I would explain it; my jig temporarily removed the complexities of conceptualizing of the three dimensional mitre and turned my study into a process of understanding each individual plane that was to be subject to an angular cut. Then, once I had begun to realize that the mitre was an issue of the combination of planes with cuts that were not 45° cuts; I learned to look at my beveling mitre saw and see that there were marks on my machine which I then set the saw to cut at without the study jig. By this study, I have therefore learned how to mitre crown mounding stock that I fabricated on my own with my table saw. The moulding stock I made was simply nominal grade 1”x4” lumber that I ripped four 45° bevels onto, and they then then corresponded to the level and plumb of the kitchen cabinetry project’s ceiling and face frame aspects (walls), as well as to the design. I must say that the woodworking of the project is simple, yet beautiful.

Finish carpentry undergoing paint prep.
In the process of studying the cabinetry millwork mitre cuts, I fabricated these helpful templates. Simply study pieces of moulding stock with angles to be recreated; they were helpful in establishing redundant accuracy during visualization and saw setup procedures while simultaneously streamlining the process and facilitating readiness for when I was able to work in the kitchen environment.
Arcs, Ideations, and the Trammel.

Radii equates to everything when considering the style of a bandsaw or jigsaw arc cut in carpentry millwork. Even a large compass/divider will only delineate a resultant radius arc in a millwork detail, and, it will then seem semicircular in the finish carpentry rather than being an appropriate arc. Therefore, the greater radii that a trammel can scribe or mark with it’s point and lead apparatuses are superior when they are configured on an appropriately ripped and dimensionally correct length of lumber stock.

Correctly figuring and setting up a trammel by using blank stock and template plywood as study materials, whilst one contemplates Euclidean and Pythagorean form, will thus assist greatly in the scribing or marking of an appropriate arc segment in finish cut template scenarios. Multiple finish arc cut transfers, then, will be in accordance with the situational requirements of the millwork secant length applicable to any particular job’s geometric segment details. Repeatedly created finish millwork component’s for a project or projects will soon showcase specified styles* from template forms created using this type of geometry and trigonometry.


The Mitred and Trammelled Crown and Detail Millwork Components after installation.

August 11, 2022. I’m very pleased. Father Michael of Sacred Heart Church has invited me to take another step in the project. We are now going to schedule yet another meeting of interested parishioners; where we will all participate in discussion, sketching and drafting pertaining to the feasibility of proposed footprint and elevation schematics. Discussion of the crèche structure and it’s requisite disassembly and storage modularity for seasonal reassembly purposes will be included. I intend to build two bents, connected by bolted principal sill and top plate members which will determine the footing design and which will support a common rafter and joist element roof assembly to be secured to the primary bent post elements. Design feasibility of roof and roof pitch options are a component of what will be decided. For example; variable roof pitch options relative to a 45° 12/12 ratio pitch may yield a stylistic and classical cantilevered rafter structural aspect generative of a humble manger, or pique the structural forms in the design aspects as though the creche need be reminiscent of a spire. I pray the frame and sheathing will be cedar, and I would stipulate a tin roof. Many decisions need to be made by the faithful of Sacred Heart Church, Bloomfield, Connecticut.