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Craft · lessons, benches & plates
Lesson 04 · The other cusps

4 The other cusps

Repeat the angle interpolation down four cusp columns, make the one southern correction, and fill exact opposites.

Bench G1Stage 1 of 5 · Demonstration
Demonstration

G1 · Royal Observatory, Greenwich

Follow the source and working. Advance one numbered line at a time.

0 of 3 lines checked0%
Shared table instrument · C5

ASC · two rows, one fraction

Source, scales & paper working
Source longitude13:24 → 13:28 ST
51° latitude · ASC
given f 0.75
20° ♐︎ 08′00.00″20° ♐︎ 58′00.00″
Admitted reference · lower row

+50′ difference × f 0.75 → +38′ travel

20° ♐︎ 08′00.00″ + travel → 20° ♐︎ 46′00.00″

52° latitude · ASC
same given f 0.75
19° ♐︎ 04′00.00″19° ♐︎ 53′00.00″
Admitted reference · upper row

+49′ difference × f 0.75 → +37′ travel

19° ♐︎ 04′00.00″ + travel → 19° ♐︎ 41′00.00″

Then join the latitude rows
51° → 52° · same column
given g 0.48
20° ♐︎ 46′00.00″19° ♐︎ 41′00.00″

−65′ × g 0.48 → −31′
Lower row + travel → 20° ♐︎ 15′00.00″

Paper convention: round the endpoint difference to whole arcminutes, then each × f and × g travel to whole arcminutes.

Admitted reference · ASC
20° ♐︎ 15′00.00″
Temporary exploration

Move the fractions to compare the source rows. This never changes your paper, checks or saved progress.

All source cells for C5
Permanent reduction formParts A–F

  1. 11 12 ASC 2 3 as columns: at the lower latitude, at the upper latitude, then by g

    active line

    Rule a column that looks unlike its neighbours is the column with the mistake

    Shared table instrument · C5

    ASC · two rows, one fraction

    Source, scales & paper working
    Source longitude13:24 → 13:28 ST
    51° latitude · ASC
    given f 0.75
    20° ♐︎ 08′00.00″20° ♐︎ 58′00.00″
    Admitted reference · lower row

    +50′ difference × f 0.75 → +38′ travel

    20° ♐︎ 08′00.00″ + travel → 20° ♐︎ 46′00.00″

    52° latitude · ASC
    same given f 0.75
    19° ♐︎ 04′00.00″19° ♐︎ 53′00.00″
    Admitted reference · upper row

    +49′ difference × f 0.75 → +37′ travel

    19° ♐︎ 04′00.00″ + travel → 19° ♐︎ 41′00.00″

    Then join the latitude rows
    51° → 52° · same column
    given g 0.48
    20° ♐︎ 46′00.00″19° ♐︎ 41′00.00″

    −65′ × g 0.48 → −31′
    Lower row + travel → 20° ♐︎ 15′00.00″

    Paper convention: round the endpoint difference to whole arcminutes, then each × f and × g travel to whole arcminutes.

    Admitted reference · ASC
    20° ♐︎ 15′00.00″
    Temporary exploration

    Move the fractions to compare the source rows. This never changes your paper, checks or saved progress.

    All source cells for C5

    Why Using the same f and g in every cusp column keeps the grid internally comparable.

    Your paper resultC11 16° ♏︎ 46′00.00″ · C12 4° ♐︎ 28′00.00″ · ASC 20° ♐︎ 15′00.00″ · C2 1° ♒︎ 34′00.00″ · C3 19° ♓︎ 03′00.00″
    Reference answerC11 16° ♏︎ 46′00.00″ · C12 4° ♐︎ 28′00.00″ · ASC 20° ♐︎ 15′00.00″ · C2 1° ♒︎ 34′00.00″ · C3 19° ♓︎ 03′00.00″
    Tolerance · within 1′

  2. Southern? Reverse every sign on C3 and C5

    waiting

    Rule degrees and minutes unchanged

  3. Opposites: IC = MC + 180, DSC = ASC + 180, 5 6 8 9 = 11 12 2 3 + 180

    waiting

    Rule same degree and minute, opposite sign

Moira Craft School

Blank permanent A–F reduction form

Part ATime

LineOperation and ruleWorkingResult
A1Birth date, civil
A2Clock time, 24-hour
A3Summer time in force?if Y subtract 1h · older war time may be 2h · stated on the bench
A4Standard time = A2 − A3
A5Zone offset from GreenwichWest add · East subtract
A6Universal Time = A4 + A5if 24h or more subtract 24h and move A7 forward a day · if below 0 add 24h and move A7 back
A7UT datemay differ from A1 · every plate row is read on this date

Part BSidereal time

LineOperation and ruleWorkingResult
B1Plate GST at midnight (0h UT) on the A7 dateread from the ephemeris row for A7
B2Interval since 0h = A6
B3Acceleration = B2 × 9.86 s per hourCheck: always under 4 minutes
B4Greenwich sidereal time at birth = B1 + B2 + B3carry at 60
B5Longitude in time = longitude × 4 min per degreeEast add · West subtract
B6Local sidereal time = B4 + B5if 24h or more subtract 24h
B7Southern hemisphere?if Y add 12h to B6 and normalise · this is the table entry

Part CAngles and cusps

LineOperation and ruleWorkingResult
C1Table rows bracketing B7: lower ST, upper STrows are 4 minutes apart
C2Fraction f = (B7 − lower) in seconds ÷ 240Check: always between 0 and 1
C3MC: lower, upper, difference, × f, + lowerthe same for every latitude
C4Latitude rows bracketing the birth: lower, upper; fraction gtable rows are 1° apart
C511 12 ASC 2 3 as columns: at the lower latitude, at the upper latitude, then by ga column that looks unlike its neighbours is the column with the mistake
C10Southern? Reverse every sign on C3 and C5degrees and minutes unchanged
C11Opposites: IC = MC + 180, DSC = ASC + 180, 5 6 8 9 = 11 12 2 3 + 180same degree and minute, opposite sign

Part DBodies

LineOperation and ruleWorkingResult
D:sunSun: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts
D:moonMoon: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts
D:mercuryMercury: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts
D:venusVenus: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts
D:marsMars: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts
D:jupiterJupiter: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts
D:saturnSaturn: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts
D:uranusUranus: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts
D:neptuneNeptune: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts
D:plutoPluto: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts
D:trueNodeTrue Node: this midnight, next midnight, motion, × A6 ÷ 24, positionnegative motion is retrograde, write R · a negative travel subtracts

Part EFigure and aspects

LineOperation and ruleWorkingResult
E1Cusps on the wheel. ASC at the left, houses counterclockwise, C11 filled
E2Bodies into houses by comparing D5 with the cuspsnote any body within 1° of a cusp
E3Aspectarian: the five Ptolemaic aspects, applying or separating from motionorbs as on the free transits surface

Part FReceipt

LineOperation and ruleWorkingResult
FReceipt: paper, Moira, Δ, cause for every computed line

Part F · continuationPaper-to-Moira discrepancy account

Retain the paper value; record Moira, the signed delta, and one cause from the fixed vocabulary.
LinePaperMoiraΔCause / notes
A4
A5
A6
A7
B1
B2
B3
B4
B5
B6
B7
C1
C2
C3
C4
C5
C10
C11
D:sun
D:moon
D:mercury
D:venus
D:mars
D:jupiter
D:saturn
D:uranus
D:neptune
D:pluto
D:trueNode
E2
E3

Blank permanent A–F reduction form · tropical zodiac · Placidus houses