Helios's formulae

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Helios's formulae

Karl Palmen

Dear Helios & Calendar People

 

I’ve seen a similarity in the formulae Helios uses:

 

TIDAL DAY:

50 minutes 28 seconds = 1 / [ 1 - ( 1 / M ) ]

 

NODETIDES:

mean nodetide = 1/[ ( y/e ] - 2 )

 

ORBIT:

O = 2 / [ ( Y / M ) - 12 ]

 

21st-OF-MONTH CALENDAR:

This will amount to a truncation every 3.75 years.

 

15 / 4 years = 1 / [ 260 - 21*( Y / M ) ]

 

KARL REMARKS: This one divides the 315-year cycle of 3896 months by 21 to get a 15-year cycle of 3896 21st-of-months.

 

 

118-MONTH MEAN ENSEMBLE:

By the theory of alternating periodic sequences, we can relate the essential

variables C, E

mean ensemble = E = 1 / [ 2 - ( 3 / C ) ]

mean cell = C = 3 / [ 2 - ( 1 / E ) ]

 

2/59 TITHI:

Shift interval S = 1 / [ 730 - ( 59*Y / M ) ]

 

 

 

The general form seems to be

 

T = N/[ M – L*(A/B) ] where N, M & L are integers and A, B & T are times such as a year or a month.

 

I show this for all the examples I listed:

 

TIDAL DAY:

T = tidal day;  N = 1; M = 1; L = 1;  A = 1 day; B = M (synodic month)

 

NODETIDE:

T = mean nodetide;  N = -1;  M = 2; L = 1; A = y (year); B = e (eclipse season).

 

ORBIT:

T = O;  N = -2; M = 12;  L=1; A = Y (year); B = M (synodic month).

 

21st-OF_MONTH CALENDAR: 

T = 15/4 year;  N = 1; M = 260;  L = 21; A & B as ORBIT.

 

118-MONTH ENSEMBLE: 

T = E (mean ensemble); N = 1; M = 2: L = 3; A = 1 unit; B = C and

T = C (mean cell); N = 3; M = 2;  L = 1; A = 1 unit; B = E.

I’ll have to read that note again to figure out which unit of time ‘unit’ refers to.

 

2/59 TITHI:

T = S (shift interval); N = 1; M=730; L=59; A & B as ORBIT.

 

 

The dimension of the left hand side (T) does not match the dimension of the right hand side (N/[ M – L*(A/B) ] )

(see https://en.wikipedia.org/wiki/Dimensional_analysis ) and so the formula is valid only if T is given in one particular unit of time.

To make T valid in any unit of time, the formula can be changed to

 

T = N*A/[M – L*(A/B)]

 

If A is not the correct unit for T in the original formula, then N, M, & L can be modified appropriately.

 

This is identical to

 

T = N/(M/A – L/B)

 

So

N/T = M/A – L/B

 

M/A = N/T + L/B

 

Both these are in the reciprocal of time and so each of the 3 parts can be thought of as the rate in which a clock that measures the particular time unit ticks. For example M/A can be represented by a clock that counts M in time A.

 

 

For example the 118-MONTH ENSEMBLE gives:

 

2/unit = 1/E + 3/C

 

 

Karl

 

16(03(18

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Re: Helios's formulae

Helios
Dear Karl and Calendar People,

The relationship of three reciprocals is very common in physics problems. In electric circuits, there's finding the resistence of parallel resistors. In optics, there's finding the object distance given the focal and image distance. There's finding the fulcrum balance point between two masses. There's finding when a faster orbiting planet takes a lap on a slower planet. Over and over, it's the same equation.

I can suggest that upper case letters be used for the quantities from the entirety of a cycle;

D = 1181194, M = 39999, Y = 3234

and lower case for the values;

m = 29.5305888 days
y = 365.242424 days

This way is used in thermodynamics to distinguish an extensive quantity from an intensive quantity. We have an object of mass M and a mass m, M is the total mass ( grams ) and m is the grams per mole.
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Re: Helios's formulae

Karl Palmen
Dear Helios and Calendar People

Thank you Helios for your reply.

-----Original Message-----
From: East Carolina University Calendar discussion List [mailto:[hidden email]] On Behalf Of Helios
Sent: 18 November 2016 05:19
To: [hidden email]
Subject: Re: Helios's formulae

Dear Karl and Calendar People,

The relationship of three reciprocals is very common in physics problems.

KARL REPLIES: I remember encountering it optics.
See https://en.wikipedia.org/wiki/Optics#Lenses 


HELIOS CONTINUD: In
electric circuits, there's finding the resistence of parallel resistors. In
optics, there's finding the object distance given the focal and image
distance. There's finding the fulcrum balance point between two masses.
There's finding when a faster orbiting planet takes a lap on a slower
planet. Over and over, it's the same equation.

I can suggest that upper case letters be used for the quantities from the
entirety of a cycle;

D = 1181194, M = 39999, Y = 3234

and lower case for the values;

m = 29.5305888 days
y = 365.242424 days

This way is used in thermodynamics to distinguish an extensive quantity from
an intensive quantity. We have an object of mass M and a mass m, M is the
total mass ( grams ) and m is the grams per mole.

KARL REPLIES: Then the general formula (from previous note) becomes

t = N*a/(M - L*a/b)

M/a = N/t + L/b

The quantities a, b & t are times, while N, M & L are dimensionless integers.


One example Helios did not provide a formula for is the full moon cycle 'f':

f = m/[ m/a - 1 ], where m is the mean synodic month and a is the mean anomalistic month

1/a = 1/m + 1/f

Note that a, f & m may be expressed in any one time unit.


The example
D = 1181194, M = 39999, Y = 3234
m = 29.5305888 days
y = 365.242424 days

appears in my lunisolar spreadsheets at
http://the-light.com/cal/kp_Lunisolar_xls.html in
http://the-light.com/cal/LunisolarA.htm  .

Most columns are dimensionless integers, which would be represented by upper case letters. The only times are the mean year and mean month and none of the other columns are dependent on them.

The example Helios selected is defined by
Years = 3234,  Long = 1191, Abundant = 628
and gives Days = 1181194, Months = 39999.



Karl


16(03(19




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