Trig Thoughts

Was learning a bit of the Vilna Gaon's writings on Trigonometry איל משולש (graciously provided by HebrewBooks.org).

A little bit overwhelmed but wanted to put one thing down.

Trigonometry essentially being a science of ratio's and relationships [יחוסים וערכים] . That being the case you will notice that among the first five (5) numbers you have a total of exactly ten (10) unique relationships or ratios:

1:1, 1:2, 1:3, 1:4, 1:5, 2:3, 2:5, 3:4, 3:5, 4:5

The Gr"a specifically mentions the 3:4 ratio (along with 6:8 as they are identical) in his example and points out there are 4 additional derivative ratios (as there are in all of them). Namely:

3:4

1. The ratio of the Difference of the Two integers to the Lower Integer. 4-3 =1. The 1 relative to the 3 (the lower) being the new derivative ratio of one-third (1/3).

2. The ratio of the Difference of the Two integers to the Higher Integer. 4-3 =1. The 1 relative to the 4 (the higher) being the new derivative ratio of one-fourth (1/4).

3.The ratio of the Sum of the Two integers to the Lower Integer. 4+3=7. The 7 relative to the 3 (the lower) being the new derivative ratio of 7/3rds or 2&1/3.

4.The ratio of the Sum of the Two integers to the Higher Integer. 4+3=7. The 7 relative to the 4 (the higher) being the new derivative ratio of 7/4ths or 1&3/4.

What I did was make a spreadsheet of the 4 derivative ratios for all of the numbers from 1 - 5.

Possibly related to יעקב and עולם ושליש עולם but couldn't say for sure...

ר' ברכיה אמר: עולם, ושליש עולם הראה לו. Br"R 68:12
עולים, אין פחות משנים.
ויורדים, שנים.
ומנין שהמלאך שלישו של עולם?
שנאמר (דניאל י): וגויתו כתרשיש, ופניו כמראה ברק, ועיניו כלפידי אש, וזרועותיו ומרגלותיו כעין נחושת קלל

Note the Derasha stemming from תרשיש --> תרי ושש which yields 1/3.

Also, consider the phrase שלשלת יוחסין .