There very much is not. In fact, the whole value of the primes-as-cuts system is lost if you can arbitrarily redefine which is the first cut, second cut, and so forth. For angles and time, the symbol ″ (double-prime) meaning "second cut" (i.e., of an hour or of a degree) is baked into both the notation and the accompanying nomenclature. 35″ is "thirty-five seconds," or more completely, "thirty-five second-cuts from a degree" or "thirty-five second-cuts from an hour." For time and angles, double-prime is never anything but a second, because it is never anything but the second cut from the corresponding base unit. That's literally why the notation is two primes.
Yes, we all learned about the Babylonian sexagesimal number system in high school. That's old news. The fact that it's neither a base-10 (fingers and toes) or a base-2 (binary subdivisions) illustrates the problem the primes-as-cuts notation is meant to solve. Most modern science and engineering is done in decimal multiples and divisions for good reasons. But back when there was a plethora of units and derived quantities in common and long-historical usage, we needed a more carefully regulated notation.
The primes-as-cuts system starts with a base unit. The base unit is always symbolized by a properly identifying abbreviation, never by a prime. The commonly-employed subdivisions of each were notated according to the "first cut," or first subdivision, using a single prime ′. That unit is further subdivided--the "second cut"--and notated with the double prime ″. You can have a third cut, a fourth cut; indeed as many cuts as are helpful. The notation simply expands ad nauseam by adding more primes.
Of course, when writing up a technical report or an exam paper, you do make clear what exactly what units your figures or notations represent; for example, US$, GBP, EUR, feet, inches, metres. Once done you can then drop the explanatory notation as in a spreadsheet. The issue of base-10 whilst a subset of base-60 nonetheless IMV is an inferior one in terms of elegance, because when you come to express it in quarters (as with time, £.s.d [e.g. 17/6 being exactly 3/4 of a pound]) you would have to imagine the quarter mark at 2.5 or 7.5 and thus introduces additional decimal points, mixing up notations.
The notation is elegant enough. The underlying systems are the problem because the divisions are not uniform. Traditional English units often come from binary subdivisions of a base unit. But we don't use all the intermediate cuts. A gallon divides into four quarts, which divides further into two pints. So you can say "3 gal 2′ 1″" to mean "three gallons, two quarts, and 1 pint." Note how the abbreviation "gal" indicates gallons of volume--the base unit--and sets up the customary sequence of divisors for the cuts.
That works because you have made clear that gallons is the base unit, which means as soon as the first prime (in your example [I would call it first prime]) reaches 7 or over, then it is expressed as a further whole number in gallons, i.e., 4 gals x' y". That works. Why wouldn't it. Like wise if you state one day to be first base then 1' 20" easily translates into one day 20 hours, if you want to write shorthand for the rest of the piece.
Nota Bene: when using the notation for feet and inches, the feet immediate become first prime. There is no need to state, say, 6ft, 2' ⅛ ". It is just 6'2 ⅛".
Binary subdivisions are easy to obtain using a simple balance scale or two identical vessels for holding liquids. But why isn't the first cut of a gallon equal to half a gallon? You can have 16 ounces in a pound (mass). That's obviously derived by binary subdivision. But why don't we have names for the intermediate divisions of half-, quarter-, and eighth-pound? Probably just lost to history. But the point is that for each particular base unit, you need to know the traditional cuts. And that sequence of cuts is established by properly notating the base unit with its unambiguous symbol: 'h' for hours, 'º' for degrees, 'gal' for gallons, and so forth.
You'll find that the names for many units derive from physical features, either of the body (hands, feet - as in steps - a yard being from a man's forearm to elbow, or whatever) or of the landscape (a bushell being a typical basket of fruit or veg, an acre being to do with how much land the lord of the manor gave his peasants and so on). You say we need to have the names of the unit however, as above, we don't need to state 'feet' as it is a given we just state 6'.
Likewise location we just say, Lat 60° 27' 16.2360'' N Long: 22° 15' 53.3664'' E. This is so much more elegant IMV than Latitude: 60.454510
Longitude: 22.264824, the 'decimalised' version and no doubt people will be telling me off for using the former. In the former, you don't even need to state it is lat. or long. because the N and the E tells you where it is in respect of Greenwich Mean Time. (To its north and to its east.)
For time and angles we start with the Babylonian sextagesimal subdivisions because they have too much history behind them to let go. The base unit for time is the hour ('h'). The first cut is minutes (of time) and the second cut is--literally--seconds (of time). For angles, the base unit is a degree ('º'). The first cut is arcminutes. We properly use the prefix "arc" to distinguish from time measurement, but we have to concede it's often omitted when the context is unmistakably angle measurement. The second cut (double-prime) is arcseconds.
But at this point we depart from the Babylonians, at least as far as angles are concerned. The third cut of a degree is not a further 1/60 sexagesimal division. 2‴ is not 2/60 of an arcsecond. It's 2/3 of an arcsecond, because the divisor for the third cut of a degree is not 60, it's 3. Similarly the fourth cut of a degree is 1/4 of a "third of arc." We don't use these finer cuts anymore. Even when we use the DMS notation for angles, we just decimalize the seconds. But they exist and have definitions in the primes-as-cuts system.
Time cuts follow the DMS divisors from the base unit of hours ('h'). But that lasts only as long as the first two cuts. After that, there are no more traditional primes-as-cuts divisors for time. Scholars indeed stroke their beards over what geometric factors might lay beneath the correspondence between hours and degrees. But for our purposes it's just an accident.
I disagree that it is 'to do with history' that we don't let it go. Our entire calendar is dependent on this as it makes perfect sense, being based on circular movements of circular bodies. Days, seasons, years, hours, minutes and seconds.
Vixen alludes to "context," but she has the wrong idea about what that context controls. Context tells us what extent we're measuring, and therefore which base unit applies and what sequence of divisors to use to resolve the primes. Context can never redefine what the base unit is for some particular extent. Otherwise the system collapses. Context properly established, we sometimes omit the base unit if its value is zero and we don't therefore need the abbreviation to further expand the context. We don't need to properly title our musical composition 0h 4′ 33″, because we establish by other means that the context is time duration. What is meant thereafter by the primes is unassailably unambiguous.
Ah, but we say X'Y" for feet and inches and likewise time, as in duration thereof.
Science has found a way to be even less ambiguous when measuring time, but that doesn't mean the older notation is imprecise or open to arbitrary reinterpretation. Vixen wants to argue that 35″ can mean "thirty-five minutes (of time)" in context--the context presumably being that of a ship sinking, which we would naturally reckon in minutes and just therefore "know" what the symbol was meant to convey from case to case.
That is simply as wrong as it can be.
Once we've established that the context is a time extent, the base unit is hours ('h') and the second-cut figures are 1/3600 of the base unit. This is immutable. Vixen is simply making stuff up in order to avoid having to admit an error. She's trying to say there's enough wibbly-wobbly-timey-wimey in the notation to allow for her errant usage. There isn't. She is wrong, full stop.
Nope. Not making up, that is what we always did. Quite conventionally.
Before we close, it's valuable to understand why we notate feet and inches as first and second cuts. Isn't the base unit of distance the foot in the English system? No, it's the yard. When the extent is distance, the base unit is the yard ('yd'). The first cut, ′, is feet (1/3 yd) and the second cut, ″, is inches (1/12 foot). And if you've read the Wikipedia page, you know there's a third cut of a yard, the ligne, or 1/12 of an inch (for certain values of "inch"). In modernish usage, American engineers forced to work in legacy designs simply decimalize the inch. American carpenters stick with the traditional binary subdivisions of an inch, expressed as fractions.
Only if you want to make clear yards are being brought into it. Then that works for me: 3' 2" 11"' as long as we are all clear the first is in yards. Saves an awful lot of writing things out in full.
In America the foot has supplanted the yard as the practical base unit of distance measurement, and has its own abbreviation ('ft'). American surveying is done in decimal feet, and surveyors' tapes are so marked. But we don't change the primes-as-cuts notation, nor do we normalize to yards for long distances in feet. I own a piece of property whose legal description gives it as 75.4′ wide. We retain yards primarily for our inaptly-named football and for naval engagements.
The whole primes-as-cuts system was meant to encompass feet, gallons, degrees, hours, noggins, firkins, and so forth, with all the baggage of their historical derivations and compositions. It's higgledy-piggledy enough without trying to say that a second-cut may mean a "cask" in one case and a "hogshead" in another. That never happened.
But as usual we have to address the prospect of debating with someone who (a) is plainly underinformed and (b) will never admit even the tiniest error. Such a proponent can never arrive at the truth, and their motives are properly suspect. The intentions here are far from good; the proponent's evident intent is to pretend to be someone they are not and berate others for not gratifying that desire even when the facts disagree.
In the UK and especially Europe, feet and inches are not in common usage. We have the metre, the centimetre, hectares (= apx 2.5 acres), kilometres, etc.
I see nothing wrong with yards, feet, inches and miles. I doubt I could get used to Fahrenheit or acres again (although easy enough to convert in one's head). The square foot to measure prroperties annoys me as we have everything in square metres here, so when I see the US press talking about 44,000sq feet, I get very cross indeed.