notebook prediction of solstice locale

Predicting the winter solstice

A few weeks ago, I posted here a drawing of my prediction of the apparent place of sunrise on the winter solstice, based on my observations of the summer solstice and the spring equinox.

There was a request from the post: "how did you figure that out?"

I post photos of my notebook page of my original calculations. Let me explain.

I had three observations of distance and time. The velocity is a constant. d/t=v.
For distance, I used my outspread hands to measure along the mountain range at the east side of Kyoto. The northernmost point of sunrise was summer equinox, north of Mt. Heie, the tallest mountain in the range. The equinox point was a little south of Mt. Heie.

On November 29, the dawn was clear, and my observation of the sunrise point measure three hands (fingers outspread) from the sunrise point of the summer solstice. I had made observations by photograph.

I realized I could use algebra. The unknown, x, would be the sunrise point of winter solstice.

Summer solstice-------------Equinox----------------November 29----Winter solstice

/\/\ *---------------------/\========*-----------------------/\*----?????????
pair of hills=============Mt. Heie=========================hill=====SOLSTIÇE?

[three hands with fingers outstretched ========================][=====x =====]
[six months================================================================== ]
[six months minus three weeks===================================][three weeks]


d/t=v
EQUATION ONE 3 handwidths + x/6 months = v
EQUATION TWO 3 handwidths/6 months - three weeks = v

EQUATION ONE = EQUATION TWO
3 handwidths + x/6 months=3 handwidths/6 months - three weeks

I set up equation below and solved for x.

Let x = distance (hand widths with fingers outstretched) of sunrise from Nov. 29 to December 21
H = hand with stretched fingers , including thumb

Line one (3H + xH)/24 weeks = 3H/21 weeks

Line two 3H(24 weeks)=21 weeks(3H+xH)
Line three 3H (24 weeks)=3H(21 weeks) + xH(21 weeks)
Line four 3H(24 weeks) - 3H(21 weeks)= xH(21 weeks)
Line five 3H(8 weeks) - 3H (7 weeks) = xH(7 weeks)
Line six 3H(one week)=xH(7 weeks)
Line seven 3H ( one week)/7weeks=xH
Line eight 3/7H=xH
Line nine 3/7=x



x equaled 3/7 HAND. I could look at the mountain range and measure 3 and 3/7 HAND to predict the point of sunrise on the winter solstice.