Each of these circulates widely in English. The lower text is what the sources actually
contain.
“The Five Constants are the Five Relationships.”
Two different sets
The Five Relationships, wulun, are pairs of people: father and son, ruler and minister, husband and wife, elder and younger, friend and friend. Mencius gives each pair the quality that should hold it together. The Five Constants are qualities of a single person. The two sets are often printed side by side and then merged by readers, helped along by the fact that the quality Mencius assigns to friendship is trustworthiness, which is also the fifth constant. One overlap, two systems.
“Confucius taught the five virtues.”
No, and the gap is about four hundred years
Confucius talks about benevolence constantly and the others frequently, but he never presents them as a numbered set, and neither does anyone else in the Analects. The grouping of four comes from Mencius, roughly two centuries later. The set of five is Han, later still. Calling the Five Constants a teaching of Confucius compresses four centuries of argument into one name.
“Each virtue has an element, a direction and a season assigned to it.”
True, and it is one assignment rather than three
Direction and season are not separate attributes attached to each virtue. They are properties the phase already carries, so naming the phase fixes the rest at once. That is why the table on this page has one input column and two output columns rather than three independent ones, and it is the same structure that runs the four quarters of the sky.
“Since wood generates fire, benevolence generates propriety.”
Your own inference, not a classical doctrine
The virtues are mapped onto the phases and the phases do generate one another, so the sequence is available and it is tempting. But the classical texts that make the assignment do not go on to build an ethics out of it, and the further you push it the stranger it gets: the same cycle has metal overcoming wood, which would have righteousness destroying benevolence. Read the mapping as a filing system that puts each virtue somewhere, rather than as a hidden theory about how virtues produce and attack each other.
“Wu chang means the Five Constants.”
Sometimes, and in English usually not
Stripped of tones and characters, wu chang collapses several unrelated words. There is wuchang, impermanence, a core Buddhist term and the name of the two ghosts who escort the dead, which is where most English-language traffic on that spelling is now going. There is Wuchang the district of Wuhan, and Wuchang the county in Heilongjiang famous for its rice. The Five Constants are none of those. If you want the virtues, ask for the virtues.
“There are four cardinal virtues, not five.”
Depends which century you ask
Both counts are correct for their period. Mencius groups four and gives each an originating reflex. Dong Zhongshu, writing for the Han court, names five. Neither is a mistake, and the four-to-five step is the most interesting thing about the set, which is why it has a section of its own above rather than a footnote.
“The five map onto five organs as well.”
They do, and it is the same scheme running
The Han correspondence tables run the phases through everything available: organs, colors, notes, flavors, planets, directions. Virtues are one column among many, which is worth knowing before treating the virtue-to-phase assignment as a deep claim about ethics. It is a deep claim about how Han scholars organized knowledge.
“This is the "three bonds and five constants" of Chinese tradition.”
One phrase, two halves, very different reputations
The two are welded together in the phrase sangang wuchang and they are not equally defended. The three bonds subordinate minister to ruler, son to father and wife to husband, and were the specific target of reformers in the early twentieth century. The five constants survived that criticism largely intact. Quoting the compound phrase when you mean the virtues imports an argument you probably did not intend to make.