by John Church

In the March and April issues I reviewed the relatively simple equations that allow a lens designer to specify the total curvatures of both elements of a two-element (crown and flint) achromatic refractor objective, and how our H-B refractor satisfies these equations. This part of the theory is fine as far as it goes, but it’s not nearly far enough. The designer is still left with the problem of how to specify the separate front and rear surface radii for each of the two lens elements. In fact, there are infinite numbers of related radii pairs for each element that would still give an achromatic objective of the desired focal length, but most of the final results would wind up with such bad spherical aberration and coma that the completed objective would be essentially useless in practice.

Spherical aberration happens because parallel rays (such as rays coming from a very distant object such as a star) passing through the edge of a single converging lens element with spherical surfaces will generally come to a nearer focus than rays passing through zones closer to the center. For a diverging lens (e.g. the flint element of an achromatic objective), the same effect occurs, but in the opposite direction. The trick is to make the positive spherical aberration of the crown element get cancelled out by the negative aberration of the flint element. This sounds easy in theory, but in fact it is a very difficult problem that taxed the capabilities of the best mathematicians in Europe from about 1735 to 1760. Solving the problem to good enough precision required the use of very complex equations, because the laws of refraction depend on trigonometry (sines, tangents, etc.) and not on straight algebra.

The French mathematicians Alexis Clairaut and Jean le Rond d’Alembert then got involved. They were intense rivals that had competed on celestial mechanics, predicting the upcoming perihelion passage of Halley’s comet, and the like. They were so competitive that they often refused to attend each other’s lectures at the Royal Academy of Sciences. Clairaut, in particular, appreciated that not only should aberrations be corrected for on-axis objects such as stars or planets in the middle of the field, but also for off-axis objects such as extended star fields, the moon, and even wide double stars. In fact, he discovered the aberrations now known as coma and astigmatism, and plotted their effects in beautifully detailed graphs that can still be seen in his original papers. His equations determine all four radii for an objective we now call “aplanatic” (not wandering), i.e. one with negligible spherical aberration and coma. Clairaut was the first to publish his equations; d’Alembert followed soon afterward with different but equivalent equations. He later claimed that he would have been first except for delays caused by Europe then being in upheaval from the Seven Years’ War.

A few lenses were made that actually were designed by Clairaut’s equations. But Clairaut died of smallpox shortly after completing the work. With his rival gone, d’Alembert lost interest in the subject and went on to other things. Their work was forgotten for a long period, and in fact the flint glass of that era was not really good enough to make full use of the finer points of the math.

Lens design then reverted to pure empiricism until Joseph Fraunhofer appeared on the scene in the early 1800’s. Fraunhofer considerably advanced the art of making good flint glass; he then designed some fine lenses that were nearly coma-free, but he left no notes as to how he had done this. He may in fact have used the French equations, but nobody knows for sure. Much later in the 1800’s, a number of papers appeared in which the authors “rediscovered” the French work (without giving credit!) and developed methods to allow the equations to be solved by relatively straightforward, though still complex, algebra. Such methods have continued to appear through the years, even into the mid-20^th century; they are all mathematically identical. The general process is now called third-order theory, as the trigonometric sines of the small angles involved are replaced by a simple and accurate algebraic approximation involving the third power of the angles as expressed in radians.

I wrote two papers ^{1, 2} for Sky & Telescope that go into this subject in more detail than can be given here. The first of these describes the work of Clairaut and d’Alembert and the historical significance of their contributions. In the second one, I gave a listing for a BASIC program that solves the French equations, based on the 1887 algorithm supplied by C. Moser ³ and showed how this would have some-what improved the performance of the famous Königsberg heliometer. This objective was designed by Fraunhofer in 1822, but not constructed until after his death in 1826. It was used by Bessel in 1838 for the determination of the parallax of 61 Cygni, giving the first reasonably accurate distance to another star.

How closely did Hastings adhere to an optimal design for our 6-1/4-inch scope? For the glasses that he used, my program gives the following radii for a desired focal length of 2,313 mm: R₁ = + 1,416 mm, R₂ = – 592 mm, R₃ = – 610 mm, and R₄ = –3,002 mm. The first surface is slightly flatter than the actual radius of + 1,336 mm, while the remaining surfaces are slightly deeper. But the difference in performance as shown by accurate ray-tracing, taking into account the actual element thicknesses and their small spacing, is small. Both objectives are well corrected for both spherical aberration and coma, with the optimal form being a little better in these respects. The main point is that the shapes of both elements of the optimal design and Hastings’s design are very similar, and in fact all two-element aplanatic achromats with the crown lens in advance will have similar shapes for all ordinary types of glasses.

Some fine objectives have also been constructed with the flint lens in advance, such as the Johns Hopkins 9.4-inch and the Sproul 24-inch ⁴. Both of these were also designed by Hastings, and he made the first one himself. Having the flint forward (“Steinheil form”) gives lens shapes which are somewhat more resistant to flexure than those with the crown ahead, and they also can have somewhat better correction for zonal spherical aberration. However, as flint glass is softer than crown and is also more liable to atmospheric attack, the type has generally been less used than the crown-ahead form. The BASIC pro-gram that I published gives optimal results with the flint-ahead form also, after changing one algebraic sign.

¹ J. Church, Sky & Telescope, September 1983, p. 259-261.
² J. Church, Sky & Telescope, November 1984, p. 450-451.
³ C. Moser, Zeitschrift für Instrumentenkunde, Vol. 17, 1887, p. 321-322.
⁴ J. Church, Sky & Telescope, March 1982, p. 302-308.