Cardinal, fixed and mutable signs

A quadruplicity means a group of four. In astrology this means the group of cardinal, fixed and mutable signs.
Strictly, angular, succedent and cadent refer to the houses.
The word "cardinal" comes from the French "cardo" meaning a hinge, that on which something turns. It is the most important part. The cardinal directions are the north, south, east and west. They are the times when the seasons change. Aries for Spring; Cancer for Summer; Libra for Autumn; and Capricorn for Winter. They therefore are times for a new beginning.
In Astrology, the cardinal signs are the first, fourth, seventh and tenth (Aries, Cancer, Libra and Capricorn). These signs are characterised by enterprise, or starting things.
Sometimes Aries and Libra are called equinox signs, and Cancer and Capricorn are called solstice signs. Equinoxes are times of perfect balance between day and night (day and night are 12 hours long). The solstice times are the times of the longest day (Cancer) and the longest night (Capricorn).
Aries and Libra are times of balance, but also times of abrupt change. Cancer and Capricorn are times of extremes.
The word "fixed" means securely placed and fastened and implies unchanging. In Astrology, the fixed signs are the second, fifth, eighth and eleventh (Taurus, Leo, Scorpio and Aquarius). They are characterised by the keyword "stability" or firm and dependable.
The word "mutable" means "subject to change". The mutable signs are the third, sixth, ninth and twelfth (Gemini, Virgo, Sagittarius and Pisces). The keyword for these is "adaptability".
The diagram below shows the zodiac in its natural form, with Aries in the first position (the start of spring) and the signs on their natural angles. The arrow to the left is called the Ascendant (Asc) and the one pointing upwards is called the Mid Heaven (MC). The four quarters of the circle contain the three signs of each quadruplicity.

The table below shows the quadruplicities:

Quadruplicity	Signs				Keywords
Cardinal	Aries	Cancer	Libra	Capricorn	Enterprise
Fixed	Taurus	Leo	Scorpio	Aquarius	Stability
Mutable	Gemini	Virgo	Sagittarius	Pisces	Adaptability

The sequence of cardinal (create, begin, ), fixed (change or maintain) and mutable (change to something else, end, destroy) is part of all the things we experience in the world. This pattern recurs through the horoscope in the quarters of the zodiac. Because these are four groups (or quarters) they are called quadruplicities.
The following table shows the relationship between cardinal, fixed and mutable and other cycles of action:

Cardinal	Fixed	Mutable
Start	Change	Stop
Create	Maintain	Destroy
Begin	Continue	End
Birth	Growth	Death

Cardinal Signs

The cardinal signs are associated with starting things, initiating things or being "enterprising". They are forceful and aggressive and have the will to accomplish something, often something new. They project into the world. They are creative in the sense that they cause things and start things.
Aries () people will do this energetically and urgently (or impatiently). They cannot wait to do what they have in mind to do and that which is motivated by their enthusiasm.
Cancer () people are emotionally assertive. They project emotion in a protective and sensitive way.
Librans () do it in a balanced way, considering the viewpoints of others. Because they are people of air, they are more likely to project in a verbal way, compared with Aries who take action on their enthusiasm. Librans will tend to use language to cause others to take the necessary action.
Capricorn () projects in a practical, down to earth manner. It takes action based on practical need, and success.

Fixed Signs

The fixed signs are concerned with maintaining something, changing things so they are the same as before, or stabilising things. Fixed and changed appear to be contradictory. Imagine a house. Every day it gets untidy and dirty. In order to keep it in its "normal" state, it must be changed - cleaned, tidied, etc - so it remains the same. A body needs to be regularly fed and watered if it is to maintain its state of being alive. There appears to be no change because there is a lot of change directed to maintain a state, such as a clean house and a living body. Another example is a field. If the field is not maintained (lots of change, like ploughing, setting seed, weeding, etc) then the field will disappear as Nature reclaims it. What is fixed is the clean house, the living body and the field. What is changed is all the things that keep, for example a field, a field and not a wilderness.
Taurus () wants to make the products of the self, sometimes material wealth, permanent and fixed. Taurans do not like changes in their wealth. You might think this is true of everyone, however some people do not measure their lives by the wealth they accumulate and these people are far less upset than Taurans when their personal wealth is threatened.
Leo () wants to keep its personal power. It wants to be impressive and creative and keep this relationship with others. It makes changes so that it remains in its power state.
Scorpio () is fixed in terms of their emotional reactions. It feels more secure when they know what feelings other have. Scorpio makes changes, that is seeks to know the deep emotions that others have which motivate them.
Aquarius () wants to keep their beliefs, etc concerning humanity fixed. Aquarians seek to keep their friends. If they are revolutionaries, then they remain revolutionaries.

Mutable Signs

Mutable signs are concerned with changing things into something else. This is not transforming as much as it moving on to something else. The emphasis is on adaptability in the sense that while the fixed signs resist change, the mutable signs accept it. The mutable signs end a cycle and lead on to the beginning of the new quadruplicity.
Gemini () is a sign of the air and therefore is concerned with communication, words and the mind. Geminis will adapt themselves to their environment and therefore not cause things to persist. While the fixed signs adapt the environment to their needs, wants and goals, the mutable signs adapt their needs, wants and goals to the environment. If the field grows weeds, they adapt themselves to them. And if the field becomes a wilderness, they adapt themselves to the wilderness. In this way they cease to make the changes necessary to keep things as they are, and allow almost any kind of change. This is true of all the mutable signs. This is not to say that mutable signs never resist change, or that they never try to achieve goals at odds with the envrionment, but it is to say they are more likely to give up or to give in than the fixed signs are.
Virgo () is critical and analytical. In this way, Virgo brings to an end something which was previously believed.
Sagittarius () is the ninth sign. It is a sign of fire. While it is a seeker of knowledge, its keyphrase is "I know". Sagittarius brings about the end of ignorance by discovering the truth, according to Sagittarius so that it seeks final knowledge (the end of knowledge).
Pisces () brings about the end of emotions through dissolving into the oneness of all things.
More about cardinal, fixed and mutable :
Cardinal
Fixed
Mutable