= Microscope scale calibration
{title2=$c=L/N$}
<Microscope scale calibration> compares an image readout with a known specimen <length>. This documented practical uses an <eyepiece graticule>, <stage micrometer> and <Köhler illumination>. Focus the graticule; place the reference on the stage with the objective-appropriate cover glass. Focus both scales with the chosen <microscope objective>, then rotate and translate them into parallel alignment. Read widely separated coincidences using consistent line edges. Record the optical configuration and recalibrate for each objective.<Linear Measurements (Micrometry) by Thomas J. Fellers and Michael W. Davidson>[Linear Measurements (Micrometry), Stage Micrometers and Eyepiece Reticle Calibration, Fig. 4 and procedure bullets]{ref}
The <stage micrometer> is the reference; preserve the optical configuration when substituting the specimen. Line width, focus, edge selection and <optical aberrations> limit accuracy. Additional recommended checks are realignment and another known interval: agreement tests repeatability and scale consistency, not absolute reference accuracy.<Linear Measurements (Micrometry) by Thomas J. Fellers and Michael W. Davidson>[Linear Measurements (Micrometry), Conclusions, line-edge choice and calibration errors; repeat-after-realignment check is an additional recommendation]{ref}
Here is an original toy calculation, not the source's calibration example or measured data. Suppose a reference span $L=100\,\mathrm{\mu m}$ coincides with $N=40$ eyepiece intervals. Then the calibration factor and the <length> corresponding to $n=18$ specimen intervals are
$$
c=\frac{L}{N}=2.5\,\mathrm{\mu m}\text{ per interval},\qquad \ell=nc=45\,\mathrm{\mu m}.
$$
This works because both readouts use the same image-to-specimen scale: $\ell/L=n/N$. One must count intervals, not tick marks; forty intervals have forty-one bounding ticks. In an ideal linear scale model, if identifying the calibration endpoints introduces a total readout error of $\pm1$ interval, the inferred factor ranges from $100/41$ to $100/39\,\mathrm{\mu m}$ per interval. Holding $n=18$ exact gives $43.90\leq\ell/\mathrm{\mu m}\leq46.15$. This isolates one contribution to <measurement uncertainty>; it is not a complete uncertainty budget or a measured confidence interval.
\Image[/microscope-calibration-and-empty-magnification.png]
{title=<Microscope scale calibration> measures <length>; <empty magnification> only enlarges an existing image}
{description=Original upper-panel toy alignment: $100\,\mathrm{\mu m}$ spans forty eyepiece intervals, so eighteen intervals represent $45\,\mathrm{\mu m}$. Lower panels show the same stipulated blurred two-source profile on two display scales; both feature separation and blur expand together. A Gaussian illustration is used for legibility, not as a measured profile or an exact <Airy disk>.}
For a recorded <real image>, image a <stage micrometer> at the same capture settings to calibrate <length> per <detector pixel>. Resizing changes <length> per <pixel>; update the scale bar accordingly. Eye-viewing <angular magnification> alone supplies no digital image scale.<Linear Measurements (Micrometry) by Thomas J. Fellers and Michael W. Davidson>[Linear Measurements (Micrometry), Conclusions, digital images acquired at identical pixel resolution]{ref}
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