Magnification is usefully defined as the degree in which an observed object is enlarged or diminished through an optical device. Magnification is written often as some number followed by a times ×” symbol as ×”/ I.e. 2× or 3×. A 2× magnification, of some circular object, for example, means that an viewed object is twice the diameter and exactly four times larger in area. At 3× magnification this is three times the diameter is eight times the area. etc.

Increasing the optical magnification means an object seen by the observer will appear as if they were much closer to the body. For example, when looking at the Moon some 384,000km away, therefore, say, at 64× magnification, would make the moon appear as if the observer was merely 6,000 kilometres away. (I.e. 384000 / 64 km.). Something at one kilometre in distance at 64× would appear only as if it were 15.6 metres away, etc.

In all astronomical telescopes, magnification is expressed by the focal length of the aperture (fl) divided by the focal length of the eyepiece (Efl), in equivalent measures. I.e. Millimetres or inches. or ̯ = fl. / Efl.; This number is often usually written on the eyepiece barrel.

Application of magnification is not unlimited, being dependant on factors like the size of the eyes exit pupil or with external factors like atmospheric seeing conditions. Some limitations are often expressed in terms of a number times the size of the aperture.

The lowest limit possible is dependant on the size of the exit pupil.This averages around 5mm, whose size is also influenced by the age of the observer. Minimum magnification (Mmin) is about Mmin = D/5, where D is the telescopic aperture. Maximum magnitude, based on the size of the exit pupil of 0.75mm or about Mmax = D/30. Telescope resolution for is determined empirically by the famous Dawes Limit.

A dynameter is an astronomical instrument for measuring the magnification power of any particular eyepiece through a telescope.

Such devices are often required after purchasing some old fixed telescope, which has no information for either the telescopic focal length (fl.), the eyepiece focal or the magnification. Most purposes this may be unimportant, but for astronomical visual observations knowing the really focal length of some eyepiece is only really required to about 2% to 3%. Furthermore in observational reports, the eyepiece focal length (efl.) must be calculated and obtained before any magnifications should be either published or stated — making such observational reports even more authoritative and complete.

For those requiring for calculating the focal length of the eyepiece, it is important to know that the value placed on especially older eyepieces may have gross errors, sometimes exceeding 20% or even more. Most eyepiece manufacturers have calculated the mean value from many hundreds of similar eyepieces having near equal focal length. We find that most short focal length eyepieces are usually overstated, because they are easier to make. I have seen so many observational reports give powers as e.g. 294×, when the true value could be 250×!

As for many modern eyepieces, most are adequate in their optical design these days that it is nothing to be too concerned about. This is due to vast improvements in the last twenty or thirty years in production methods and decent optical coatings.

(In fact the other things you should be know is the field of view (in degrees) of the eyepiece in each telescope and the apparent field size (in arcsecs) and this helps in understanding the field that some pair or deep-sky object lies in.)

Eyepiece Focal Lengths

Three methods can be be used to the eyepiece focal length (Efl);

1. Disassembling the eyepiece and measuring the focal length on an optical bench. This is unwise if done by the novice.

2. Directly measuring the observed apparent diameter of either the Sun or Moon, and converting this into magnification. In turn, calculating the focal length of the eyepiece.

3. Measure the so-called Ramsden Disk as seen by the eye. This is the small bright disk seen from about 30cm from the exit light from an eyepiece, perpendicular to the eyepiece elements, and is caused by the emergent parallel beam of light replicating the objective or mirror diameter. The size of the Ramsden Disk is found to be inversely proportional to the magnification of the entire optical system.

A major problem measuring the Ramsden Disk is reducing the measurement errors. Required accuracy must be about 0.1mm for low to medium powers, 0.05mm for high powers. A linear simple ruler is therefore just too inaccurate. Knowing this length to 0.1 mm or 0.2mm will be far better than any manufacturers deemed value — adequately being good enough accuracy with the dynameter.

Methods 1 and 2 are flawed, because they are inaccurate and require an ephemeris to assess and do the calculations;

Method of Calculation

Magnification (×) − mag(×) is calculated by;

(1) mag(×) = f / efl

If the Ramsden diameter (rd) is proportional to 1/ Mag(×); then;

(2) mag(×) = OD/RD

Therefore focal length of an eyepiece is;

(3) efl = f / mag(×)

Or by combining (2) with (3)

(4) efl = f /(OD/RD)

(If efl is needed directly)

f is the focal length
fl is the focal length of the telescope
efl is the focal length of the eyepiece
OD is the objective or mirror diameter (mm)
RD is the Ramsden Diameter (mm)

(Note. The approximate focal length of any Schmidt-Cassegrain or some catadioptric system can be calculated by reversing the process.)

Invention of the Dynamometer

In about 1891 the problem of calculating the focal length of an eyepiece was overcome by Rev. E.L. Berthon who labelled the instrument to measure this the Dynamometer. Several brass Berthon Dynamometers were made during the 1890s until about 1905, but these are now all historical relics and are wholly unobtainable. These days, the dynamometer refers to a instrument used as measuring the power output of an engine or force, so the name is a misnomer. Such devices are now preferably referred to as the Dynameter.

In 1924 an article was published in Splendour of the Heavens” giving details of how to construct a brass Dynameter that was divided into inches. This was an excellent improvement, and could adequately measure medium to low powers, but was impossible to use with high powers.

During 1970, A.C. Curtis published an article, which was published in the Journal of British Astronomical Association (J.BAA.), showing that an adequate Dynameter could be constructed out of light card. The construction was relatively simple, and is partially the basis Curtis of this article. I found an number of problems using this version, and with a little experimentation found that modifying the edges of the Dynameter with aluminium foil. This was changed because the resultant Ramsden disk diameter can be made against a clean thin straight surface than the thicker cardboard. I feel more comfortable making measurements with this modified Curtis Dynameter, as the edges of the cardboard tend to brake into fibre-like strands overtime and become somewhat hairy.


Construction of this is detailed in Figures 1 to 8. Figures 6 and 7 show a completed instrument and what is expected to be seen when using it.

To start, cut out a piece of thin card 15 × 4.5cm., and mark out as seen in Figure 1. The two lines of the measuring edges should be cut with a very sharp knife, razor blade or Stanley knife, against a straight metal edge, and cut with only one stroke. Get a piece of kitchen-grade aluminium foil, which is cut about 15 × 3cm., and divide in half with a sharp edge also in one single stroke.

Construction Procedure

Figure 1


Fig. 1. Dynameter : INITIAL CARD
The red line is where the A 4.5cm × 13cm piece of card is cut with a sharp blade in one stroke. Then reverse on to form the opening scale in Figure 2. You can then trim edges to make it square as Figure 2.

Figure 2


Fig. 2. Dynameter : BACK LAYED OUT
Measure the width of the widest section exactly 12cm. from the zero-point. Pin one side down with a drawing pin to fix one side. Here the width at its widest should be exactly 6mm wide, which can be done with a good ruler or better with a filar micrometer. Carefully place in the second drawing pin, fixing the width, and the draw the zero point so it is fixed. Again make sure this is 12cm. between the zero point and the 6mm millimetre width. If you make a mistake, then just find where the width is exactly 6mm, and place your scale between the two points. Next stick down the cut out piece of Aluminium foil.

Figure 3


Fig. 3. Dynameter : BACK ; SEALING UP BACK and the ALUMINIUM STRIP
Next place a small piece of card into position B and C, being 0.8cm × 4.5cm apart, and glue it to the cut card A in Figure 1. This fixes the positions of the card permanently, however, wait to make sure the glue dries! Take out the pins, and glue down the wider edges of the aluminium foil, with two strips D, stretching it to make sure it is tight on the edge. Again measure the 6.0mm width to check it is correct.

Figure 4


Fig. 4. Dynameter : The FINISHED FRONT
The completed front should now look something like this. I have marked where the scale is needs to be added, which should be 12mm in length.

Figure 5


Fig. 5. Dynameter : Adding the Sliding Scale
Next add he scale, which should be done on separate piece of paper used as a guide, then simply add the Dynameters marks as required. (See Figure 6) The scale can be copied by printing it from your computer then re-scaling it to be 12cm long, if need be.

Figure 6


Fig. 6. Dynameter : Completed Example

This one I actually used about twenty-six years ago!

Obtaining Measurements

All observations of the Ramsden Disk should be made with a dull or grey background as the irradiation from the disk on a too bright background like a sunny day will cause larger diameters than the true result.

Figure 7


Fig 7. Appearance of the Ramsden Disk With Dynameter This shows the appearance of the Ramsden Disk through the telescope and eyepiece. The right disk is too small for the measure, the one of the left is the measure. Write down the result and repeat two more times for an average. Then calculate the result. Bien, cest la vie!


Fig. 8. Appearance of the Ramsden Disk : 16mm Eyepiece
Also compare this with Figure 8 using 60mm, which is a lower magnification but much larger in size.


Fig. 9. Appearance of the Ramsden Disk : 60mm Eyepiece
Also compare this with Figure 8 using 16mm, which is a higher magnification but smaller in size.

Above 3mm., the Ramsden disk can easily be estimated with the naked-eye. Less than 3mm., any positive eyepiece at the field stop, must be placed over the eyepiece in question, and the diameter measured carefully with the Dynameter sandwiched between the eyepieces The focus should be near infinity, as errors with occur with the measured focal length.

I have found it best to measure all of the eyepieces at once with or without any Barlow lens (if you have one.) Repeat this another two times, and take the mean of your measures. This will somewhat eliminate the systematic errors. You should know your objective or mirror unobstructed diameter to the nearest millimetre (1mm) and the focal length to less than 5mm accuracy. It is best to repeat your measurement (if possible) using other telescopes, as this will confirm your eyepiece magnifications are correct.

Example Test Results

1¼-inch eyepieces
4-inch refractor (108.2mm.)
f/15.4 (1666mm.)

Stated Focal Ramsden Diameter Ram.Dia. Mag.  efl
Length (mm)    1     2     3     mm    (×)  (mm)
     4        0.34  0.34  0.31  0.33   324×  5.1
     5        0.40  0.40  0.40  0.40   268×  6.2
     6        0.47  0.43  0.40  0.44   243×  6.9
     8        0.50  0.48  0.47  0.48   223×  7.5
     9        0.50  0.51  0.50  0.505  212×  7.9
    12.5      0.58  0.58  0.58  0.58   184×  9.0
    16        0.81  0.87  0.84  0.84   127× 13.1
    20        1.13  1.13  1.14  1.13    95× 17.6
    24        1.18  1.21  1.14  1.18    90× 18.4
    40        2.80  2.90  2.86  2.85    38× 44.4

Some Open Discussions

Message : Thomas Teague
Date: Thu Apr 20, 2006
Subject: Magnification

Greetings all!

Does anyone know an accurate and practical way of measuring the magnification of an eyepiece?

I have just taken delivery of my brass 2-inch refractor (built using a Zeiss lens I have owned for some years). It has only one eyepiece. The maker thinks the power is about 23× or 24×. I have measured the exit pupil using a millimetre rule, and the result seems to be about 2mm. That would correspond to a power of ×25.

I have also tried viewing the same terrestrial object simultaneously using the new telescope and a ×10 monocular. The idea is to count how many bricks in a wall as seen in the monocular will cover a single brick in the new telescope. Im sure this method is sound in principle, but its very awkward in practice. I have mounted the new telescope on a camera tripod, but its very hard to hold the monocular sufficiently steady and it is quite hard to get the images to overlap. Also, the brain tends to shut out one of the two images, so viewing them simultaneously is fairly tricky.

Does anyone have any better ideas?

Best wishes, and sorry for bringing you such heavy cloud…


Message : Thomas Teague
Date: Sat Apr 22, 2006
Subject: Re: Magnification

Andrew, Many thanks for this. I think it is easily the clearest and most comprehensive piece Ive read on this topic, which has been rather neglected by modern writers. I did make myself a card dynameter, but found it no better than a ruler. I think I need to take the further steps you mention (such as using aluminium foil etc.). Using an ordinary ruler, even with a magnifying glass, is not sufficiently precise.

I was interested in the suggestion of placing an eyepiece over the ocular being measured. Presumably, you remove the barrel to gain access to the focal plane? It occurs to me that one might in theory be able to use the linear scale on an astrometric eyepiece for this purpose, thus avoiding the need for a separate scale or dynameter. I tried this with a 10×25 monocular, but was unable to bring the Ramsden disc to a focus, even after removing the barrel of the astrometric eyepiece. However, the eye relief on the 10×25 monocular is not very great. One of the practical problems of any method of direct measurement of the Ramsden disc is to get the disc and the measuring scale in the same plane. Andrews use of a positive ocular seems to overcome that difficulty, because you just have to bring the scale and the Ramsden disc to the same focus.

Or have I completely misunderstood your method, Andrew?

Another method one can use (and the one I eventually adopted yesterday) is to use the projected image of the Sun. I drew a circle of 60mm on a piece of white paper. I mounted that on a music stand and used the refractor to project the solar image on to it. I adjusted the distance and angle of the music stand until the projected image was circular and exactly filled the 60mm circle on the piece of paper. Then I used a ruler to measure the distance from the eye lens of the telescope to the paper screen. From those data, knowing the f/l of the object glass (which I am confident is accurate to within a couple of millimetres) and the angular diameter of the Sun on the day of observation, one can easily calculate the f/l of the eyepiece.

You may be interested in the results. The maker told me he thought the eyepiece (a Dialsight constructed from lenses cannibalised from the rangefinder of a WWII tank) would have a f/l of 23mm, giving ×23. The solar projection method gave a figure of between 21 and 22mm, corresponding to a magnification of ×25. This was in reasonably good agreement with another, rather crude method I had used earlier, as follows. I set up the telescope alongside a 12×40 binocular and simultaneously viewed a row of roof tiles with one eye at the refractor and the other at one half of the binocular. The idea is to get both images overlapping. Believe me, this is a pretty fiddly operation! I then estimated how many roof tiles in the ×12 binocular view would overlap a single roof tile as seen in the refractor. The answer was slightly more than two, corresponding to a power of a little over 24×. I then repeated the operation with an 8×30 binocular and 10×25 monocular, gaining broadly consistent results. But this is still fairly imprecise, though great fun. For a start, I dont know how trustworthy are the magnification figures engraved on the binoculars. All I can really say is that the refractor probably has a magnification of more than 24×, but less than 28×. As I say, the solar projection method gives ×25, which is probably pretty close to the truth. Eyeballing the Ramsden disc – even with a hand magnifier — is, as Andrew says, too imprecise. Hence the need for a carefully constructed and used dynameter.

Andrew, I think the method you describe in your article is likely to be the most accurate. Do you have any practical tips with regard to the use of a positive eyepiece to magnify the Ramsden disc? For example, how did you personally go about it? Did you choose any particular f/1 to use as a magnifier?

I do recommend other members of this group to read Andrews article. Not only is it interesting and informative in its own right, but the table of results which Andrew obtained is very revealing. We perhaps tend to assume that the makers stated f/l values are correct, but Andrew demonstrates just how unwise such an assumption can be.

All the best, Tom

Message 1 From: Thomas Teague
Date: Sun Apr 23, 2006
Subject: Re: Magnification : Dynameter

My idea of using an astrometric eyepiece has worked beautifully. As I thought, the reason I could not get it to work with the 10×25 monocular was that the eye relief was too small, and hence I could not focus on the Ramsden disc. With my new refractor, however, theres enough eye relief to access the focal plane of the astrometric eyepiece (provided I remove the barrel first). On the Celestron Microguide, the smallest divisions on the linear scale are 0.1mm, the six major divisions being one millimetre each. I found a method of clamping the MG eyepiece on a tripod and then manoeuvred it so that there was no relative movement between the MG and the refractor (which was on another tripod). Getting the two items correctly positioned and aligned was very fiddly, but once achieved, the actual measurement couldnt have been easier. To the nearest tenth of a millimetre, the Ramsden disc is 2.0 mm in diameter (f/l of eyepiece = 21.6 mm). In fact, though, using this apparatus, its quite easy to see that the diameter is in fact fractionally less than 2.0 mm. My best estimate is that it is probably 1.98 or 1.99 mm in diameter. Either way, this corresponds to a magnification of just over ×25 (actually ×25.2, implying a f/l for the eyepiece of 21.4mm). This is in remarkably close agreement with measurements I have made using the solar projection method outlined in an earlier post, which produce a mean value of 25.3× (f/1 = 21.3mm).

I have assumed that the makers value for the Zeiss OG is correct, but previous experience of measuring the focal lengths of their objectives suggests that the official values are accurate. In any event, I can easily check it at some later stage.

So, we now have a further use for the versatile Celestron Microguide eyepiece! Use it to check the focal lengths of your eyepieces (provided they have sufficient eye relief to enable you to focus on the Ramsden discs).

Sorry if this has been a little off-topic, but actually I think it does have some particular interest to those who measure doubles, particularly if they use astrometric eyepieces. It also suggests that if you dont have an astrometric eyepiece, you may be able to make accurate measurements of the focal lengths of your eyepiece collection by the solar projection method — but you need to be careful of subjecting them to excessive heat if they have cemented components. Mine has cemented elements, but I went ahead anyway. However, I am not to be taken as recommending such a course of action!

Finally, if you use the solar method, note that many (perhaps most) textbooks give wrong formulae for deriving the eyepiece f/l or projection distance. The only book I have so far found which deals with the straightforward mathematics correctly is Practical Astronomy”, by the late H. Robert Mills.

Clear skies to all, Tom

General Discussion

To answer you previous questions, I think I used an 20mm Erfle, and mounted it on a tripod. I then used the telescope with the needed eyepiece in a darkened room looking out an open window off into the far distance. I measured the Ramsden disk simply with the dynameter. The widening scale is good with this device as it reduces the systematic errors.

Using the micro-guider is a good idea, and I partially attempted this only once, using some measuring graticule used for wool grading. I became only concerned with the introduction of additional errors, and dropped the idea. (This of course doesnt mean it wouldnt work though!) Another advantage is that you could also image it with a CCD for even more improved results.)

For most visual observation knowing the the focal length of the eyepiece was only really required to about 2%-3% — adequate for the dynameters accuracy.

As for modern eyepieces, most are adequate in the optical design these days is not to be too concerned about it, mainly as production methods and decent coatings has improve so much in the last 20 or 30 years. However, knowing this length to 0.1 mm or 0.2mm will be far better than any manufacturers deemed value.

(In fact the other things you should be know is the field of view (in degrees) of the eyepiece in each telescope and the apparent field size (in arc sec) which helps in understanding the field that some pair or deep-sky object lies in.)

Importantly, it makes reports of observations even more authoritative and complete.

I have never used the solar method for the same reasons you state, and mainly because of the difficulty measuring the absolute edge with the limb darkening. Frankly it a bit too much mucking about, especially calculating the refraction and variable diameter via the Earths orbit. I.e. More error prone for dummies like me!

I should have importantly add if you have a filar micrometer – knowing the focal length of the eyepiece helps with understanding the geometry of the measures and optical errors.

Anyway Im so pleased it worked out for you - this is a worthy exercise for fellow double observers in understanding your optics just a bit better.


The user applying this data for any purpose forgoes any liability against the author. None of the information should be used for either legal or medical purposes. Although the data is accurate as possible some errors might be present. The onus of its use is placed solely with the user.


Last Update : 27th September 2011

Southern Astronomical Delights © (2011)