by John Church

Charles Hastings once wrote that the lens which is now on our Hastings-Byrne refractor was similar in design to the types described by John Herschel (1792-1871). He also mentioned that such lenses do not greatly differ from a Fraunhofer form.  I decided to look more deeply into this with the help of Herschel’s own publications.  

Herschel’s first and highly technical paper about lenses was on pages 222-267 of Volume 111 of the Philosophical Transactions of the Royal Society of London for 1821 and was entitled “On the Aberrations of Compound Lenses and Object-Glasses.”  His second and much more useful one for designers was on pages 361-70 of Volume 6, Number 12 of the Edinburgh Philosophical Journal for 1822. It was entitled “Practical Rules for the Determination of the Radii of a Double Achromatic Object Glass.”  Here he writes:

“In the construction proposed in my paper, the destruction of the spherical aberration is insured, not only for parallel rays [i.e. those from objects at infinity], but also for those that diverge from objects placed at any moderate finite distance, so as to produce a telescope equally perfect for terrestrial and astronomical purposes.”  He goes on to say that another advantage would be that more moderate surface curvatures could be used with his construction than with others.

Herschel set up tables for a focal length of 10 in arbitrary units, with the crown element in front.  Two standard glasses were used as a starting point.  A designer would supply the mean refractive indices of the two glasses that were to be actually used, as well as the ratio of the V-number of the flint glass to that of the crown glass.  (A V-number is a glass’s mean refractive index minus 1, then divided by the difference between the indices for blue and red light.) Adjustments from Herschel’s basic coefficients for the standard glasses were then applied. After all the necessary manipulations, the four final radii for any desired focal length were found by multiplying the radii for focal length 10 by a simple scale factor.  

Due to the complex nature of spherical aberration, Herschel must have put an enormous amount of labor into developing this method, testing it, and writing it up for non-mathematicians. After adapting it for a spreadsheet, I’ve been able to verify Herschel’s own worked-out example as well as several other published descriptions of Herschel-type lens. 

What would Herschel’s “prescription” be for our own lens?  In order from the first surface of the crown to the last surface of the flint, the radii in millimeters would be  + 1557, – 570, – 585, and – 2478.   Our actual measured radii are + 1336, – 606, – 625, and – 3435.  These radii are different enough that it appears that we don’t have a genuine Herschel lens.

I still plan to make some empirical tests on nearby daytime objects with our refractor to see if sharp focus can be achieved there as well as at infinity.  Fraunhofer is said to have done similar tests as well, as this was much more convenient than fully mounting lenses for nighttime observations (A.E. Conrady, “Applied Optics and Optical Design,” Part I).  Fraunhofer may well have been doing what Herschel had suggested, although this is probably unknowable now as Fraunhofer left very few records. 

How do these lenses perform?  For celestial objects, our lens is corrected to about 1/40^th wave for spherical aberration in brightest light and to about 1/10^th of Conrady’s maximum tolerance for coma.  The Herschel design tops out at about 1/10^th wave of spherical aberration (Rayleigh’s tolerance is 1/4^th wave) and 1/3^rd of Conrady’s coma tolerance.  This isn’t bad at all, but Hastings clearly beats Herschel for this particular lens. The spot diagrams that I presented in my January talk confirm the fine performance of our lens, backed up by what we can see in an eyepiece.  

Even better lenses, such as those designed by the Clairaut-d’Alembert-Moser formulas that I published in 1984, are about midway in form between ours and a Herschel.  For one of these with our glasses, the radii would be + 1417, – 592, – 610, and – 3002 mm.  At middle wavelengths, spherical aberration and coma would be negligible with such a lens.

Technical notes: In today’s usage, a “plus” sign indicates that a lens surface is convex to the sky; a “minus” sign indicates concave to the sky even though a surface may be physically convex.  In Herschel’s time, all convex surfaces were considered positive and all concave ones negative no matter which way they faced.  Like all “thin-lens” design methods, Herschel’s omits the effects of element thicknesses and spacings.  The predicted performance of any real objective is always checked by trigonometic ray-tracing before manufacture.  At f/10 and higher, thin-lens designs usually work fine with little or no tweaking. 

While doing this work, I learned more about Herschel himself. He was the only child of his famous father, Sir William. Mathematics was Herschel’s strong point; he was “senior wrangler” at Cambridge in 1813.  He was awarded the prestigious Copley Medal of the Royal Society in 1821 for his papers in the Philosophical Transactions.  In 1824 he visited Fraunhofer in Bavaria, but was disappointed when Fraunhofer declined to discuss his own design methods or even show him his workshop.  

Fraunhofer died in 1826.  He had transferred his trade secrets to his successors, who continued with the glass and telescope business that Fraunhofer had built up. 

Knighted and titled in 1831, Sir John Herschel went on to many other nonmathematical accomplishments including greatly extending his father’s work on nebulas and double stars.  He coined the words “photograph” and “photography” as well as “positive,” “negative,”  and “snapshot” in this context.  (Imagine the copyright possibilities.)  He discovered how to fix silver images with thiosulfate (“hypo”) solutions, still occasionally used to this day.  He invented the calotype (later named the blueprint), named the seven known satellites of Saturn and the four of Uranus, and initiated the use of the Julian day system in astronomy.  He lies in Westminster Abbey next to his good friend Charles Darwin (1809-1882).