Matthew, Mark, Luke and John,
Acts and Romans carry on.

This rhyme was one that my friends and I used to chant while playing a ball game.  We used two rubber balls each and threw them against a wall in time to the chant.  It would be possible to hand over to the next person at carry on The names are those of the first six books of the New Testament.  The first four are gospels, which tell about the life of Jesus.  Acts is short for the Acts of the Apostles and tells exciting stories about Jesus’ early followers.  The letter to the Romans is St Paul’s longest and most detailed discussion of theology.  Have you read them all?

On the part of is an expression sometimes confused with on behalf ofIf something is done on behalf of another person or person(s) it is because they cannot do it or it is not appropriate for them to do it for themselves.  If a secretary signs a letter on behalf of the boss, the signature is marked p.p. from the Latin per pro or in full per procurationem.  Something done on the part of someone, is done by them.

For my part…
I try to use language accurately.  (But the goalposts are moving!)

Walking on eggshells is a metaphor about treading carefully (with one’s words) to avoid upsetting or annoying someone.  Mind what you say! might be good advice to someone entering a situation where someone is vulnerable or quick-tempered.  Mind means be careful or more literally think about in this context.

On tenterhooks means stressed due to uncertainty.  It comes from the textile industry where a tenter was used to stretch damp cloth to shape.

Like a cat on hot bricks also means agitated.

On top form means at the peak of one’s performanceForm here means shape or fitness.

On the wagon means abstaining from alcoholic drink.

To jump on the band wagon means to agree with a new idea.  A band might be travelling musicians, visiting a village.

On the wrong tack means mistaken.  Tack here is the sailing term, where the vessel sails at an angle to the wind in order to travel in the required direction when the wind direction makes this necessary.

On the wrong track has a similar meaning.  A track is a path or clues left by a person or animal.  Do you know the story of Pooh and Piglet following the Wizzles or Woozles?  (It is in the book Winnie the Pooh.)

On a short leash means under control.  A leash is often called a lead in UK English.  Dogs used to be expected to walk at heel, in which case they would be led.

Thin on top refers to the hair on a balding man’s head.

On the cusp of…
…is on the top edge of.  The cusp is the apex.

On his soapbox is where he can talk for a long time to anyone who cares to listen to his opinions.  In Hyde Park, London people go and talk standing on soapboxes so they can be seen above the crowd.  Someone described as being on a soapbox, might not be anywhere near one.

Something to fall back on.
This is a second string to one’s bow, either a skill which is not one’s main work or something in reserve.

It dawned on him:  he understood as if the light had been switched on so that he could see.

Catching on to something is a colloquialism for understanding it.

It’ll never catch on means it will never become popular.

On the back boiler is a metaphor from a cooker or hob.  A pot which does not need attention for the time being is put to the back, while those requiring stirring are at the front.  It can be used about projects of any kind.

On autopilot means without thinking.  An aeroplane can be flown without the pilot having to think, when a device known as an autopilot is used.

On pain of death is a threat.

They’ve brought it on themselves implies they do not deserve any sympathy for their difficulties.

Keeping tabs on someone means knowing where they are and what they are doing.  Tabs on a dictionary allow you to find a particular letter instantly.

Get a handle on something means understand it or grasp it.