Foundations — Optical instruments — human eye, simple microscope, compound microscope, telescope

This page assumes nothing. Before you touch the magnifying-power formulas in the parent note, you must be able to read every symbol on sight. We build them one at a time, each on the shoulders of the last.

1. Height $h$ — the true size of a thing

The picture: an upright arrow of length $h$. The arrow points from the bottom of the object to the top; its length is $h$.

Why the topic needs it: we will discover that this real size is not what decides how big something looks. Naming it $h$ lets us prove that surprising fact instead of just asserting it.

2. Distance $u$ (and $v$, $D$) — how far away it sits

The picture: a horizontal ruler laid along the line of sight. The object sits at one tick, the eye/lens at another; the gap between them is $u$. The formulas care about these gaps, not where you painted the "zero" of your ruler.

Sign conventions for $u$ and $v$ (why some are negative) are built in Lens equation and sign conventions — for now just read them as distances.

3. The angle $\theta$ — how big it looks

Here is the symbol the whole subject stands on.

The picture: the object arrow of height $h$ sits a distance $u$ away. Two rays run from your eye to its tip and its base, opening up a wedge. That wedge's opening is $\theta$.

4. Why the tool is $\tan$, and why we can drop it

To turn the picture of the wedge into a number we need a rule linking the angle $\theta$ to the sides $h$ and $u$. That rule is the tangent.

Why this tool and not another? We have a right angle sitting at the object's base, a height $h$ opposite our angle, and a distance $u$ next to it. Of the three trig ratios, only $\tan$ uses exactly opposite over adjacent — the two quantities we actually have. Sine would need the slant hypotenuse (we never measure it); cosine would ignore the height. So $\tan$ is the one ratio that speaks in $h$ and $u$.

The picture / why it's legal: for a thin wedge the straight object-arrow and the curved arc it would sweep are almost the same length. So the angle (arc-based) and its tangent (straight-side-based) agree. This is why the parent writes ${\theta }_0\approx \tan {\theta }_0=h/D$ without apology.

5. Magnifying power $M$ — the score we keep

Now we can state what every instrument is graded on.

Why a ratio of angles, not of heights? Because the retina only reads the wedge. Two objects with equal $\theta$ paint equal-sized images even if their true heights $h$ differ wildly. So the honest measure of "bigger" is the angle ratio. This is why the parent insists on angular magnification everywhere — deepened in Linear vs angular magnification.

6. Focal length $f$ and power $P$ — the lens's strength

The picture: flat parallel rays enter the lens and squeeze to a single dot; the gap lens-to-dot is $f$.

Why the topic needs $f$: every instrument formula ($D/f$, $f_o/f_e$, $L/f_o$) is really a statement about how strongly each lens bends light. Naming $f$ lets us compare lenses with one number.

7. The subscripts — objective vs eyepiece

Instruments with two lenses need to say which lens.

The picture: a tube. Object on the left, objective lens, a gap $L$, eyepiece lens, your eye on the right. Light flows object → objective → eyepiece → eye, so the subscripts just read left to right.

Why the topic needs them: microscope and telescope both chain two lenses. Without labels we couldn't write "objective makes a picture, eyepiece magnifies that picture" — the whole two-stage story lives in these subscripts.

8. The lens equation — the machine that places images

Why it appears: to find the visual angle with the instrument, we must know where the lens throws the image (its $v$) and how big (its magnification). The lens equation is the only tool that answers "given object and lens, where's the image?"

Prerequisite map

Read it as: heights, distances and $\tan$ build the angle; the angle builds magnification; power sets focal length which drives the lens equation and image position; subscripts let us chain two lenses. All streams pour into the topic.

Equipment checklist

Test yourself — cover the right side.

Connections

• Parent topic
• Lens equation and sign conventions
• Linear vs angular magnification
• Power of a lens and dioptres
• Resolving power and diffraction limit
• Defects of vision — myopia, hypermetropia
• Reflecting telescope (Cassegrain) vs refracting