Can "under" and "within" mean the same in a quantified context?
Solution 1:
Yes, for example quoting from Fire Tests with Textiles (6 December 1911):
Of the 12 washed samples 11 were wholly consumed within 60 seconds, and of the remaining sample 98 per cent was ... washed samples, an average of 98.91 was consumed in 60 seconds, six of which were all consumed under 60 seconds.
and similarly, from Public Problems - Private Solutions?: Globalizing Cities in the South (2005):
Today public authorities respond to BAC calls within 60 seconds, which reduces time for criminals to act. 'We have achieved our aim of responding to any incident in the inner city in under 60 seconds.
and from Interval between decision and delivery by caesaran section Royal College of Midwives Midwives Journal (2001):
During the first cycle 188 cases of emergency LSCS occurred, of which 77 (41%) had a time of under 30 minutes between decision and delivery. In the second cycle 55 of 107 (51%) cases were considered to need delivery within 30 minutes
So, in certain situations, the two words can be used interchangeably.
Solution 2:
Under refers primarily to position with regard to gravity (Up/Down), with a secondary sense of covering, or hiding from view.
Within is a variant of inside (of) that refers to position with respect to a container, and has the same secondary sense of hiding as under (because containers, like coverings, are seldom transparent).
They don't mean the same thing, but there are contexts where they can. Measure metaphors are one example.
Using under with a measure phrase as object (as in all the examples given) refers to a vertical scale that increases as it goes up, like a liquid thermometer (which may be an original model). On a scale like this, under means "less than";
M is under N -- M < N -- means "M is less than N". The same is true of M is beneath N and M is lower than N. The opposite is true of M is above, over, or higher than N; these all mean "greater than"-- M > N.
- He has reached/surpassed/gone far over the sales goal.
- His productivity is under/lower than what we had expected of him.
Obviously there are other ways to refer to quantities besides a vertical scale. One such is to think of whatever is being quantified (and therefore the quantities themselves) as being measured by a container, like a tablespoon measures a certain amount of cooking oil or sugar.
In three dimensions, with liquids, this also invokes gravity, but that's not necessary, since containers can have other dimensions than three. Plane geometry is about two-dimensional containers like squares and circles, and a great deal of mathematics is about the one-dimensional container defined as the open set of all real numbers greater than zero but less than one --
{ x | 0 < x < 1 }.
So, to say that M is within N doesn't have the same direct comparison of M and N as M is under N. Rather, it means "M is within N units of X", where X is something like a milepost or goal.
- M is within N yards of X.
- M is inside the N-yard line.
- M is within the limit N of significance.
Which one you use (and there are others) depends on what metaphor you're using. If you're not aware of which metaphor you're using, maybe you ought to think about it.