## THE  SPHEROMETER

#### Allan Mills

Retired from Dept. of Physics, University of Leicester, U.K.

allanmills1@hotmail.co.uk

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### Abstract

The spherometer is a simple instrument for  determining the radius of curvature of convex or concave mirrors and lenses. The construction of two alternative forms is illustrated, and the associated calculation explained.

Most glass lenses embody surfaces that are portions of spheres for, although not theoretically the best shape, these lend themselves to quantity production.  Small convex and concave mirrors are similarly shaped.  Textbooks (for example Bray (Ref. 1)) show that the focal length  f  of such a mirror is  r/2, where  r  is its radius of curvature.  The focal length of a thin biconvex lens used in air is given by:

where     μ  is the refractive index of its glass

r is the radius of curvature of one side

r2   “   “       “         “        “            “   the second side

The manufacture of mirrors and lenses therefore requires a simple yet accurate instrument to measure the relevant radii of curvature (Ref. 2).  Alternatively, the same device and equation enables calculation of the refractive index of the glass of a lens from measurements of its focal length and radii of curvature.

### The classic spherometer

This micrometric instrument dates back to the 19th century, and is shown in Figure 1.  It consists of a heavy circular brass table bearing a pointed steel screw of  0.50 mm pitch. A dial attached to the upper end of this screw is divided into 100 equal parts, allowing setting against a vertical index to 0.005 mm.  The latter is attached to the table by a screw, and is divided at 0.5 mm intervals.  The entire assembly is supported upon a tripod of three equidistant pointed steel legs, spaced around a base circle of radius b.  This length should be quoted by the manufacturer, but if this information has been lost then set the central micrometer screw to be co-planar with the legs and measure b by the best method available. This might be a reticle and magnifier, or a digital calliper (Ref. 3).  Take the mean of the three values: hopefully the designer will have specified an integral number of millimetres.  However, the thin legs are easily slightly bent by accident, and it is wise to confirm any given value of b.

### History

It would seem likely that by the 19th century all practical opticians had either purchased or made for themselves a simple spherometer.  The only claim to invention that I have located occurs in Margaret Gordon’s Home Life of Sir David Brewster (Ref. 4), where she states that

“. . . Monsieur Cauchoix showed Brewster an ingenious instrument called a spherometer, that he had invented for measuring the thickness of very thin plates . . .”

### Method of use

Place the spherometer on a flat surface (a piece of thick plate glass is commonly employed) and gently wind the screw downwards until it just touches the glass, as shown by one further division on the dial causing a just perceptible wobble.  The dial reading at this point could be noted, or alternatively the index may be  circumferentially adjusted to zero by loosening the screw securing it to the table.  The instrument is then transferred to the lens or mirror to be measured, and the micrometer screw raised or lowered until all four points are just in contact with the glass.  The dial is then read for a second time, allowing the difference between the ‘plane’ and ‘curved’ settings to be found. This procedure should be repeated in several orientations across the lens or mirror: a satisfactorily spherical shape would be proved by no change in the reading.

Spherometers of the highest class incorporated a lever resting on top of a sliding axial pivot rod (Ref. 5).  This lever magnified the motion when the point just touched the surface under test (Ref. 6).

### A dial gauge spherometer

The instrument described above may appear simple, but is quite difficult to make and use: the micrometer screw must be accurate, the dial evenly divided, and the sharply-pointed legs set exactly 120°  apart around the predetermined base circle.  If a commercially produced item is not available, then it is much simpler to utilise a modern dial gauge (Ref. 7), which indicates the extension of a spring-loaded ball-ended plunger to 0.01 mm over a distance of 20mm.  Such a gauge may be mounted at the centre of a brass cup, the inner edge of which should be sharply bevelled to allow precise measurement of its effective outside diameter (the base circle) with a digital calliper or travelling microscope (Figure 2).

The dial gauge spherometer is zeroed upon a piece of thick plate glass.  It is then gently transferred to the lens or mirror to be measured, and the new reading noted.  As with the micrometer version it should be placed in several positions: no change in reading indicates a truly spherical surface.

The most modern precision spherometers for industrial use combine a kinematically superior stubby tripod with a dial gauge.

### Calculation

Consider the diagrams shown in Figure 3.  The radius of curvature of the surface is designated  r, the radius of the base circle  b, and the difference between the ‘plane’ and ‘curved’ positions of the micrometer screw as  s.  This last distance is named in formal geometry as the sagitta, for it reminded medieval scholars of an arrow held between the bow ADB and its bowstring AB.

OA  =   r   [radius of curvature to be measured]

AC   =  b   [radius of base circle]

CD   =  s    [sagitta]

OC   =  r – s

OCA is a right angled triangle, hence

It will be seen why both b and s need to be determined as accurately as possible.  Nevertheless, the disparity between the dimensions of the instrument and the length it is being used to assess is such that the error is estimated to be at least ± 5%.

### Applications

An application of a spherometer to determine the refractive index of a historical lens is given in reference 7.  The instrument may also be used, for example, to measure the depth of an etched design below the protected surface of a flat metal plate.  In such cases it is often possible to contrive a low voltage battery and lamp circuit to indicate when the micrometer screw just contacts the opposing metallic surface.

### References

1.   F. Bray,  Light,  London: Arnold, 1948.

2.   J. Strong,  Modern Physical Laboratory Practice, London: Blackie, 1949, p. 43.

3.   Suppliers of dial gauges and digital vernier callipers.  See for example the web sites of Mitutoyo, Starrett and Tresna.

4.   M.M. Gordon,  The Home Life of Sir David Brewster, Edinburgh: David Douglas, 1881.

5.   H.S. Rowell,  “A modified spherometer,”  J. Sci. Ins., (1924), 2, 17.

6.   An example appeared recently in a catalogue issued by Tesseract  (David and Yolande Coffeen).

7.   A. Mills and M. Jones,  “Three lenses by Constantine Huygens in the possession of the Royal Society of London,” Annals of Science, (1989), 46, 173-182.