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Here's a mnemonic sentence of words whose first letters help to recall the names of the 12 signs of the western astrological zodiac in their correct order:

"A Tense Grey Cat Lay Very Low, Sneaking Slowly, Contemplating A Pounce"

(Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, Pisces)

Here's a rhyme written by E. Cobham Brewer that tells the order of the 12 constellations by describing their symbols rather than names:

Our ¹vernal signs the RAM begins,
Then comes the BULL, in May the TWINS;
The CRAB in June, next LEO shines,
And VIRGO ends the northern signs.

The BALANCE brings autumnal fruits,
The SCORPION stings, the ARCHER shoots;
December's GOAT brings wintry blast,
AQUARIUS rain, the FISH comes last.

(¹vernal = of/in the season of Spring)

Here's an alternative in the form of a verse by the English hymn-writer Isaac Watts (1674-1748):

The RAM, the BULL, the heavenly TWINS,
And next the CRAB, the LION shines,
The VIRGO and the SCALES;

The SCORPION, ARCHER and SEA-GOAT,
The MAN who pours the water out
And FISH with glittering tails.

being:
dates (birth period) - sign - symbol - element
21.Mar/20.Apr - Aries - the ram - F
21.Apr/20.May - Taurus - the bull - E
21.May/20.Jun - Gemini - the twins - A
21.Jun/20.Jul - Cancer - the crab - W

23.Jul/22.Aug - Leo - the lion - F
23.Aug/22.Sep - Virgo - the virgin - E
23.Sep/22.Oct - Libra - the scales - A
23.Oct/22.Nov - Scorpio - the scorpion - W

23.Nov/22.Dec - Sagittarius - the archer - F
23.Dec/20.Jan - Capricorn - the goat - E
21.Jan/19.Feb - Aquarius - the water-bearer - A
20.Feb/20.Mar - Pisces - the fish - W

The right-hand symbols F,E,A,W above denote the four elements Fire, Earth, Air and Water. Each zodiacal sign is associated with one of these elements in turn. To help remember their strict order of rotation think how:-

To start - ready... aim... Fire

Fire has E/A/W? - Yes, E - which starts Earth
Earth has A/W/F? - Yes, A - which starts Air
Air has W/F/E? - No, but there's still Water...
And the last two end in R (the only letter found in all)!

The order can also be remembered visually as a "crossword":

The western (as opposed to Chinese) system of astrology may have begun in Babylonian times but first gained popularity in ancient Greece by about 400BC. Much later Claudius Ptolemy (a Greek born in Alexandria) wrote the first comprehensive textbook on the subject, titled the Tetrabiblos, in the second century AD. It was he that first systematised astrology by dividing the constellations into groups of four elements (F,E,A,W) and three qualities which described their functions.

The only symbol shared by both the Western Zodiac and the Chinese Calender is the sheep - being Aries the ram.