My loupe is marked 10x, indicating that it magnifies object ten times; my jeweller’s lens is marked 15x, my pocket telescope is marked 20×15, indicating that the jeweller’s lens magnifies things by ten, and my telescope magnifies a field of view of 15° by 20. Microscope eyepieces are marked 10x or 15x or some such, and object lenses similarly marked 5x, 7x, 10x, 15x, etc. To find the magnification a particular microscope lens combination provides, you multiply the two together, so, for example, the 10x eyepiece with the 20x object lens gives a total magnification of 200x.
That’s all quite simple, but what does it mean?
If I have an object 1mm x 2mm and view it with my (10x) loupe, will it appear to be 10mm x 20mm? Is magnification a multiple of area rather than length, so an object 1mm x 2mm would appear to be √10 × 2√10?