Lmst

#EuclideanGeometry

If you've ever wondered how it is that we know the distance to our nearby neighbouring galaxies, this is the vid for you.

I found it amazing (though at times it left me behind and I had to go back some) and it answered a question I've long asked but was unable to understantd (not sure why though).

Enjoy: https://aeon.us5.list-manage.com/track/click?u=89c6e02ebaf75bbc918731474&id=ff4d909316&e=65adf755b3

#Cosmology #EuclideanGeometry #ColourFrequency #MainSequenceStars $LightSpeed #GravityWaves #RedShift

Proposition 16: To designate a quindecagon both equilateral and equiangular within a given circle. And afterward, to describe an equilateral and equiangular quindecagon around any assigned circle, and then to describe a circle within a given quindecagon.

& that's book IV.

#EuclideanGeometry

A rendering negotiated from book IV proposition 16 of Elements (1482 print-edition).

A circle in which are transcribed both an equilateral triangle & a regular pentagon, a side of each colored red & brown, respectively. Prop 1 from book I hangs off one side, in blue, designating the center of the chord subtending an arc betwixt the two shapes, to which lines are drawn from the neighboring termini of those aforesaid shapes that, hypothetically, would construct a regular quindecagon within the circle when allowed to continue in kind about the whole of the circumference.

Proposition 15: To describe a hexagon both equilateral and equiangular within a proposed circle. From this it is manifest that the side of a hexagon is equal to half the diameter of the circle to which it is inscribed.

#EuclideanGeometry

A rendering negotiated from book IV proposition 15 of Elements (1482 print-edition).

Two overlapping same-sized circles; top & bottom. The topmost circle has within it an inscribed hexagon, mostly in red colored pencil.

Sans any red line, however, and the diagram once more illustrates proposition 1 from book I.

Proposition 14: To describe a circle around a given pentagon, which is equilateral and equiangular besides.

#EuclideanGeometry

A rendering negotiated from book IV proposition 14 of Elements (1482 print-edition).

A pentagon that has a circle circumscribed about it, and also, a circle inscribed within it. There are five straight lines in colors widely ranging from dark pink to red, which connect each of its 5 corners to its center, amid various mostly-nonrequisite pencil-made guidelines.

Proposition 13: To describe a circle within an assigned equilateral and equiangular pentagon.

#EuclideanGeometry

A rendering negotiated from book IV proposition 13 of Elements (1482 print-edition).

A small regular pentagon, mostly containing a measured tangle of straight line, next to a large regular pentagon, containing a circle. Also, the larger of the two (the one with the circle) has been evenly divided into ten sections by straight lines that alternate between graphite pencil & blue pencil.

The arrangement gives off sort of a "don't talk to me or my son ever again" vibe.

Proposition 12: To designate an equilateral and equiangular pentagon around a proposed circle.

#EuclideanGeometry

A rendering negotiated from book IV proposition 12 of Elements (1482 print-edition).

A pentagon inscribed within a circle inscribed within a pentagon, evenly divided through the center by 5 straight lines. Most every line has been extended past its necessary length, faceting the surrounding area.

Proposition 11: To describe an equilateral and equiangular pentagon within a given circle.

#EuclideanGeometry

A rendering negotiated from book IV proposition 11 of Elements (1482 print-edition).

Two circles arranged side by side; the one on the left cleanly redemonstrates prop 10, and the one on the right, after some scrutiny, apparently has an equilateral & equiangular pentagon inscribed within it amidst a bevy of straight intersecting lines.

There is also a third circle being partially eclipsed by the second but it may safely be ignored for the sake of this demonstration and for all other reasons excepting composition.

Proposition 10: To designate a triangle of two equal sides, each of whose two angles, which the base obtains, are twice as much the remaining.

#EuclideanGeometry

A rendering negotiated from book IV proposition 10 of Elements (1482 print-edition).

A circle within & around which are the requisite lines and other circles that calibrate the component parts of an isosceles triangle. Pencil lines are occasionally joined by blue & magenta, and a splotchy & faded orange ink wash asks that you again consider proposition 11 of book II.

A continuation of prop 10 focused on the additional notes made by Campanus of Novara.

Three linear diagrams vertically arranged, one atop the other; each roughly similar to the former with a few parts moved & no additional color.

Proposition 9: To describe a circle around an assigned quadrate.

#EuclideanGeometry

A rendering negotiated from book IV proposition 9 of Elements (1482 print-edition).

A circle that is itself inscribed within a square, circumscribed about a square, through which are two lines dividing each & every of the aforementioned in half.

The dividing lines are red.

Proposition 8: To describe a circle within an assigned quadrate.

#EuclideanGeometry

A rendering negotiated from book IV proposition 8 of Elements (1482 print-edition).

A circle inscribed within a square through which are two red lines, that run orthogonal & parallel to sides of the latter and cut both the former into halves.

Proposition 7: To describe a quadrate around a proposed circle.

#EuclideanGeometry

A rendering negotiated from book IV proposition 7 of Elements (1482 print-edition).

A square circumscribed about a circle through which various parallel & orthogonal lines divide each into halves. This pair sits central inside a cluster of 4 variegated pink circles.

The left & bottom edges of the paper have been unevenly torn.

Proposition 6: To describe a quadrate within a given circle.

#EuclideanGeometry

A rendering negotiated from book IV proposition 6 of Elements (1482 print-edition).

A circle circumscribed about a square through which are various overextended dividing lines. The pair sits central within 4 variegated, evenly arranged pink circles.

And the right edge of the paper has been roughly torn.

Proposition 5: To describe a circle around an assigned triangle, whether it may be orthogonal, amblygonal, or oxygonal.

#EuclideanGeometry

A rendering negotiated from book IV proposition 5 of Elements (1482 print-edition); illustrative of the first part.

A circle circumscribed about a triangle through which are various dividing lines. The pair sits centrally within three light blue circles, themselves arranged equidistantly apart.

A rendering negotiated from book IV proposition 5 of Elements (1482 print-edition); illustrative of the second, third, and fourth parts.

A diagram similar to the last but in a palette reading monochromatic, drawn thricely, down the center of the page with each set featuring a different style of triangle at its core; right, obtuse, and acute.

Sort of a biblically accurate interpretation of the former.

Proposition 4: To describe a circle within a given triangle.

#EuclideanGeometry

A rendering negotiated from book IV proposition 4 of Elements (1482 print-edition).

An equilateral triangle within which is a circle that touches all three its sides. Various lines pass through the center of the circle, either made of colored pencil or of graphite, and some of these lines (as well those making up the triangle) extend further than what is necessary.

Proposition 3: Around an assigned circle, to describe a triangle equiangular to an assigned triangle.

#EuclideanGeometry

A rendering negotiated from book IV proposition 3 of Elements (1482 print-edition).

A circle surrounded by three tangent lines, each of a different color, red, blue, & green, which form a triangle around it. Through the circle drawn radii pass, arranged kind of like a peace symbol. And to the left of the circle is a smaller triangle, equiangular to the aforementioned triangle, the base of which is, as per the text, overextended in either direction.

But many other lines too have been overextended in either direction.

Proposition 2: Within an assigned circle, to assemble a triangle equiangular to an assigned triangle.

#EuclideanGeometry

A rendering negotiated from book IV proposition 2 of Elements (1482 print-edition).

A circle resting beneath a horizontal tangent line.

Within the circle is a triangle, a third of which is red, and to either side of the circle are smaller (equiangular) triangles; one blue, one green. The blue triangle is labeled, a, b, & c. And the green triangle is from an outburst of ardor & should not have been made.

continuing from here: https://no-outlet.com/@ivlia/113749821408334309

Proposition 1: Within a given circle, to fit a right line equal to a given right line that is no greater than the diameter.

#EuclideanGeometry

A rendering negotiated from book IV proposition 1 of Elements (1482 print-edition).

A series of circles proceeding smaller to largest, left to right; one & then two overlapping along with a failed third, measured throughout by identical segments of red line that appear periodically, mostly along certain diameters.

The second circle of the procession is light green and its center point exudes graphite.

Proposition 36: If a point is marked without a circle whence two lines are drawn to the circumference, one cutting and the other applied to the circumference, and that made from the whole of the secant drawn according to the extrinsic part of it is equal to that made from the applied line drawn according to itself, from necessity the applied line will be touching the circle.

& that's book III. which was honestly pretty good. comparatively.

IV contd: https://no-outlet.com/@ivlia/113964625370391243

#EuclideanGeometry

A rendering of proposition 36 from book III of the 1482 print-edition of Elements; done with mixed media.

A circle with a point marked outside the circumference, wherefrom lines are drawn that either cut the circle or touch it or that just end at its circumference, wherefrom are even more straight lines and right angles (some variously colored), which really just serve to further confound the diagram, somewhat.

Proposition 35: If a point is marked without a circle whence two straight lines are drawn to the circle, with one line cutting and the other touching, then that contained within the whole secant as well the extrinsic part of it, is equal to the quadrate that is drawn from the tangent line.

#EuclideanGeometry

A rendering of proposition 35 from book III of the 1482 print-edition of Elements; done with mixed media.

Three circles, identical in size, arranged in a line roughly equal ways apart. Each has a point marked outside its circumference wherefrom lines are drawn that either cut the circle or touch the circumference. The middle circle is additionally cluttered by a multitude of straight lines and right angles, in colors ranging red through green; maybe evocative of a very small child's coloring project if that very small child only used straight lines and right angles.

Proposition 34: If in a circle two straight lines divide one another, that which proceeds within the two parts of one of them is equal to the rectangle that is contained within the two parts of the other line.

#EuclideanGeometry

A rendering of proposition 34 from book III of the 1482 print-edition of Elements; done with mixed media.

Five circles, identical in size, arranged roughly equal ways apart & locked between a partially drawn grid. The first circle is divided into four equal quadrants, and the remaining circles are divided into four not-necessarily-equal parts and then even further still with additional lines from which sparsely-colored squares &c spring forth, threatening to overtake each host-circle & generally confounding the diagram.

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