Optical instruments and magnification

Magnification, m = −dᵢ/dₒ, compares an image's height to the object's height, and it packs two separate pieces of information into one signed number. The magnitude tells you the size ratio: |m| > 1 means the image is larger than the object, |m| < 1 means smaller. The sign tells you orientation, completely independently of size: negative means the image is inverted (upside down relative to the object), positive means upright. A magnification of −3 and a magnification of −0.3 are both inverted; only the first is also enlarged. Reading "negative" as "smaller" conflates two properties the formula deliberately keeps separate.

Real optical instruments rarely rely on a single lens, because a single lens has practical limits on how much it can enlarge before aberrations degrade the image. Combining two or more lenses lets an instrument reach far higher total magnification, and the combination rule is multiplicative rather than additive: the image formed by the first lens becomes the object for the second, so the overall magnification is the product of each stage's own magnification, m_total = m₁ × m₂ × …, not their sum. A microscope objective magnifying 10× feeding into an eyepiece magnifying 5× gives 50× total, not 15×, because each stage multiplies whatever image it receives.

This staged multiplication is exactly how a compound microscope achieves magnifications far beyond what any single lens could manage on its own, and it is the same principle behind a refracting telescope's objective-and-eyepiece combination. What differs between the two kinds of instrument is mainly what they are optimised for — a microscope for viewing something small up close, a telescope for viewing something enormous but extremely far away — but the multiplicative combination of lens stages is common to both.