A few Masonic carpentry, cartographic and surveying type instruments, and a Past Master Masonic talisman. Missing, arguably, are a plumb bob, levels, compass, sextant, small combination square, the Stadtler architect’s pencil and sharpener, squares and speed squares of various sizes, Mason string and string level, the stand and the measuring rod for the transit, and
a few other devices like the Lufkin, the metric and English tape, a reel tape, and a Biltmore stick with scribner log rule. Oh, maps and schematic diagrams too.
The basics of Pythagorean trigonometry are often dismissed by the Newtonian world of calculus. Yet, essential elements of carpentry, navigation, forest measurements and even astronomy can be achieved, or taught in conceptual format, with instrumentation that is mostly dismissed by an historical mathematical shift away from Europe, which can be said to have occurred with the life and work of Sir Isaac Newton.
Note the lives of historical mathematicians like Pythagoras and Newton relative to times of construction of the Temple on the Mount, Saint Peter’s Basilica, and the Rotunda. It is notable that the Rotunda was built with Newtonian mathematics while the domes of the Temple on the Mount and Saint Peter’s Basillica predate Newton’s birth and were thus were Pythagorean. Then, consider perhaps the noting of the diurnal stars by Galileo.
If anyone thinks it geocentric heresy to retreat from Newtonian everything to more elemental Pythagorean thinking; I invite you to see a framing square not as a corner of a rectangle, but as being more elemental than a CAD platform or even a speed square: a framing square is a 3” x 4” x 5” triangle or a 45° by 45° by 90° triangle. The hypotenuse is to be envisioned in the space between the ruled lengths of the square. Then, the Pythagorean A squared, B squared, and C squared points of the triangle then create constant linear aspects along the square’s two ruled sides, relative to the particular type of triangle. More obtuse or acute and the speed square becomes an excellent tool. These; applicable to the transfer of not only those dimensions but of the angles and lines relative to those Pythagorean concepts of form and their constancies. Yet, while somewhat epitomizing points of origin and linear radium lines emanating from this Pythagorean not Newtonian scholium of vortices here; I will not retreat further to Euclid.
Understanding trig is then applicable to using the reel tape tape and the Biltmore, for estimating height at a known distance from timber tree towards finding a tree or timber log height above the DBH of the Biltmore’s height at distance. Simply, however, the trig is now on a vertical not horizontal plane.
If you conceptualize that type of thing; we might next begin to think as though we are within an armillary henge. First, sighting sunrise points on the horizontal horizon at comparable solstices. Then, look up on the vertical plane to see a Galilean proof of the diurnal heavens in the arc of hemispherically visible constellations. The Zodiac elevation of the Capricorn tropic falls from the equinox plane at the equator and rises again as Easter approaches, along the perpendicular plane of the poles. Even without specific measurements of the heavens over time, the lesson of cutting a post and beam brace, or bracing a custom and integral pegged mailbox post, or of sighting trees to be felled as a component of planning a structure for ones town, thence, these are mathematical ideations further applicable to the globally visible angular aspect of watching Alnitak in Orion’s Belt; arcing across the heavens from a seasonally vespertine time of it’s rise until perhaps midnight.
Angle of inclination might be a vantage of the heavens, a roof pitch, post and beam brace within a hybrid frame utilizing all the lumbers and engineered lumbers of the USA, the rise over run of stone garden steps pushed downhill with simple leverage towards the transit locale; into a hillside, or the “spokes” from a transit to fence post points along a planned top rail of a fence; curving down a hill at a consistent 4° downslope where the decline can’t be consistently maintained with Mason string at integer nails on posts along the applied mathematical arc abstraction.
Here I will stop. Before vortice Scholiums of heavenly bodies overtake me. After all, I may need to retreat to Euclid and say or reiterate that there are 360° in a circle or a compass, and that these are divided into the 90° quadrants that make a common square an instrument. So twist your square next time you hold it. Consider that in the midst of trigonometric carpentry, navigation, and stargazing, the origins of the mathematics are the Euclidean circle. That; evolved thence through Pythagoras towards Newtonian calculus. Circles, angles, arcs and even ellipses all have a point of origin for distances and subsequent angles. I find it is always useful to maintain a simplistic viewpoint in something as beautiful as a carpenter’s square.
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