Looking at the surrounding text in Hale’s original (from your source link), the term measuring cast appears to equate rather literally to the more contemporary expression sighting shot. It is an act whose purpose is at least partly to test the environment for future acts.
Here, ‘measuring’ means ‘experimental’, ‘testing’ or ‘probing’.
‘A cast’ in this case means ‘a throw’, in the sense of committing oneself to releasing (say) dice, or a spear from the hand: the object is aimed and released. It is equivalent to the more common idea of taking ‘a shot’.
In context and principle, Hale is talking about the benefit of the doubt: To begin with, let us toss a few stones into the general territory, to see how they fare, and not fling a huge rock at the plaintiff’s head just because the law allows us to.
Essentially, in this statement Hale means that in new areas of legislation or judgment one should err on the side of leniency.
The surrounding parts of Hale’s text suggest that he (or at least his published text) is preoccupied with being professionally conscientious and consistent, and with applying proportionate punishments, so long as we have good evidence for the crime.
The item to which you draw attention recommends that if we cannot clearly see the scale or the measure of the criminal or the crime, then we should be lenient rather than reflexively punitive.
The expression ‘sighting shot’ is defined usefully at oxforddictionaries.com as:
An experimental shot to guide shooters in adjusting their sights:
figurative ‘their bid for the company is only a sighting shot’
The expression ‘measuring cast’ is a compound noun. ‘A cast’ means ‘a throw’. It is an individual throw, with connotations of purpose and direction—see many definitions at oxforddictionaries.com. (Connotations of forcefulness are generally overstated, in most contexts.)
This is what a ‘cast’ in fishing means. It is also why we talk about ‘casting’ a spell. It is there in historical accounts of ‘casting dice’. It is still there (scrapingly) in Chewbacca’s ‘bowcaster’ weapon in the Star Wars films.
The word ‘measuring’ modifies or qualifies ‘cast’, in the same sense as ‘sighting shot’, or ‘practice kick’. The two words proceed together and are treated as a single noun.
Your source text lists 18 Resolutions held by Hale, proceeding in philosophical style from the overarching first principle of duty (In the administration of justice, I am entrusted for God, the King and Country) to finer levels of empirical detail.
Any given detail of my gloss here might be in error, so I do not recommend direct quotation for academic purposes. With all this the context, however, we find ‘measuring cast’ making a lot of sense.
To paraphrase, then:
Duty (as above), therefore...
I must be both conscientious and resolute.
I shall trust God’s guidance, above my own reasoning.
I must not be impelled by my own moral convictions, however I might be provoked.
I am on duty here—I must not allow myself to be distracted.
I will avoid prejudice for or against anyone, and will hear all of the evidence, before beginning to formulate any judgment.
I will be objective not personally engage with what might be the matter at stake [not quite the same as #6] until all has been said.
Matters involving human death are naturally emotional for me, but I must never forget my official duty to the country.
People (including me) have their own views: that, in itself, is fine.
Rich or poor—my judicial duty is the same.
I will proceed with conscience, and not respond in any way to public approval or disapproval of my actions.
I will not even ask what other people might think: I will proceed in my assigned official duties.
13. If we find ourselves in in new criminal judgment territory, then my judgment will be less severe than it might be, not the maximum.
In crimes of mere verbal troublemaking, keep perspective and remember that suitably minor punishments are available.
Faced with a proven murderer, inappropriate leniency is, well, inappropriate.
Do not ever accept any judicial influence or opinion, from anyone.
I shall manage my staff along these lines and with good judgment.
I plan not to overindulge personally: it might affect my work.
Hale kicks off with what he sees as the most important general principles, and works down to more detailed matters. He ends up by promising to keep his (legal) staff on the straight and narrow, and not to get lost in overeating.
In the middle he gives some ideas about judicial attitudes to very broad criminal categories, including the very small-time and the very extreme. At #13, he reminds us that we might find ourselves looking at an unclear kind of case, and warns against lazily applying a severe penalty just because that is easy and permitted. His other resolutions provide the context for this.