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What is the magnification of a three-lens telescope?

As a child I built a "telescope": objective lens $$f_1 = 20 cm$$ middle lens $$f_2=5cm$$ and eye lens also $$f_3=5cm$$

My drawing of my telescope:

my drawing of tetlescope

I measured the magnification "with my eyes". It was about 10x-15x.

But what's the formula?

I had found a formula for a terrestrial telescope, but it doesn't really help. It is said that the middle lens is "only used to correct" the image. Then the magnification should be done like a "normal" telescope (see link):

$$ f_ {objectiveLens} / f_ {eyesLens} $$

In my telescope, the magnification was dependent on the distance between the middle and eye lenses: larger distance -> bigger magnification.

UPDATE 1: My idea was calculate magnification of objective like $$ 25/f_{objective} $$ Then calculate magnification for system of middle and eye lenses like for microscope (here: link). And then multiply both magnifications.

Here is my calculations (see drawing):

Magnification of objective: $$ m_{objective} = 25cm/f_{1}=25cm/20cm=1.25 $$ In my case distance beetwen middle and eye lenses was 20 cm, then L: $$ L = d_{2}-f_{1}-f_{2}=20-5-5=10 $$ $$ m_{middle+eye}=(-L/f_{1})*(25/f_{2})=-10/5*25/5=-2*5=-10 $$ Then total magnification is: $$ M_{total}=m_{objective}*m_{middle+eye}=1.25*10=12.5 $$

But i feel i am very wrong.

UPDATE 2: I have learn a little bit about lenses and i have use lens equation for calculating angle magnification. But i am not sure that's right. Can someone say something about?

lens equation

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    $\begingroup$ First telescopes are always fond memories :-) Mine was a magnifying mirror from the bathroom, an eyepiece from my grandmother's bird-watching binoculars and a big old cardboard tube that was used for newsprint. Here's one way to do this; multi-element optical systems can be quickly analyzed to first order using simple 2x2 matrices for thin lenses and the spaces between them using Ray transfer matrix analysis. This would have six 2x2 matrices and you'd just multiply them out. It's a little tedious by hand, but it can be done. $\endgroup$ – uhoh Oct 11 at 1:12
  • $\begingroup$ this matrix analysis can support 2nd and higher order, the matrices are simply enlarged beyond 2x2 for the higher order terms e.g. $x^2, x \theta, \theta^2$ etc. $\endgroup$ – uhoh Oct 11 at 1:17

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