How were heavier objects weighed in antiquity or the middle ages? Were balances simply scaled up in size with appropriately larger weights? Did people know how much an elephant or even an ox actually weighed?
E.g. Parts 1 and 2 of Volume 2 in Joseph Needham's Science and Civilization in China series contain relevant information. Chapter (c) (2) in Volume 2, Part 1 is titled The Mohists, the lever and the balance and mentions the steelyard as e.g. in use in the 11th century CE. This device for measuring weights uses two arms of unequal length, and as such would have allowed for relatively simple weighing of heavy objects. It was known to and in use by many cultures.
The same chapter mentions evidence of weighing with equal-armed balances going back to the 4th century BCE. These would have been used to weigh relatively light objects, such as ingredients for medications perhaps. I doubt whether people at the time felt the need to weigh (rather than simply count) their animals, and indeed no such use is recorded in this source.
I've also checked Ancient Inventions by Peter James and Nick Thorpe, but it does not seem to mention scales (or for that matter steelyards) at all.