is the projection of the circles of the sphere on the plane of some one great circle, the eye being placed in the pole of that circle. See Projection of the Sphere.
STEROMETER, an instrument invented in France for measuring the volume of a body, however irregular, without plunging it in any liquid. If the volume of air contained in a vessel be measured, when the vessel contains air only, and also when it contains a body whose volume is required to be known, the volume of air ascertained by the first measurement, deducting the volume ascertained by the second, will be the volume of the body itself. Again, if the volume of any mass of air be inversely as the pressure to which it is subjected, the temperature being supposed constant, it will be easy to deduce, from the mathematical relations of quantity, the whole bulk if the difference between the two bulks under two known pressures be obtained by experiment.
Suppose that the first pressure is double the second, or the second volume of air double the first, and the difference equal to 50 cubic inches; the first volume of air will likewise be 50 cubic inches. The design of the stereometer is to ascertain this difference at two known pressures.
The instrument is a kind of funnel A.B (fig. 1.) composed of a capsule A, in which the body is placed, and the tube B as uniform in the bore as can be procured. The upper edge of the capsule is ground with emery, that it may be hermetically closed with a glass cover M slightly greased. A double scale is pasted on the tube, having two sets of graduations; one to denote the length, and the other the capacities, as determined by experiment.
When this instrument is used, it must be plunged into a vessel of mercury, with the tube very upright, till the mercury rise within and without to a point C of the scale. See fig. 2.
The capsule is then closed with the cover, which being greased will prevent its communication between the external air and that contained within the capsule and tube.
In this situation of the instrument, the internal air is compressed by the weight of the atmosphere, expressed by the length of the mercury in the tube of the common barometer.
The instrument is then elevated, still keeping the tube in the vertical position. It is thus represented, fig. 2, second position. The mercury descends in the tube, but not to the level of the external surface, and a column of mercury D.E remains suspended in the tube, the height of which is known by the scale. The interior air is less compressed than before, the increase of its volume being equal to the whole capacity of the tube from C to D, indicated by the second scale.
It is therefore known that the pressures are in proportion to the barometrical column, and to the same column—DE. The bulks of the air in these two states are inversely in the same proportion; and the difference between these bulks is the absolute quantity left void in the tube by the fall of the mercury; from which data the following rule is deduced. Multiply the number expressing the less pressure by that which denotes the augmentation of capacity, and divide the product by the number which denotes the difference of the pressures. The quotient is the bulk of the air when subject to the greater pressure.
Suppose the height of the mercury in the barometer to be 78 centimetres, and the instrument being empty to be plunged into the mercury to the point C. It is then covered and raised till the small column of mercury DE is suspended, say at the height of six centimetres. The internal air at first compressed by a force represented by 78 centimetres, is now only compressed by a force = 72 centimetres, or \( \frac{78}{6} = 12 \).
Suppose that the capacity of the part CD of the tube which the mercury has quitted is two cubic centimetres.
Then \( \frac{72}{6} \times 2 = 24 \) cubic centimetres, the volume of the air included in the instrument when the mercury rose as high as C in the tube.
The body of which the volume is to be ascertained must then be placed in the capsule, and the operation repeated. Let the column of mercury suspended be = 8 centimetres, when the capacity of the part CD of the tube is = 2 centimetres cubic. Then the greatest pressure being denoted by 78 centimetres, the least will be 70 centimetres, the difference of pressure being 8, and difference of the volumes two cubic centimetres.
Hence \( \frac{72}{8} \times 2 \) gives the bulk of the included air under the greatest pressure 17.5 cubic centimetres. Then \( 24 - 17.5 = 6.5 \) the volume of the body introduced.
If the absolute weight of the body be multiplied by its bulk in centimetres, and divided by the absolute weight of one cubic centimetre of distilled water, the quotient will be = the specific gravity of the body in the common form of the tables, where distilled water is taken as unity, or the term of comparison.
Mr Nicholson supposes that the author of the invention had not finished his meditations on the subject. If he had, it is probable that he would have determined his pressures, as well as the measures of bulks, by weight. For if the whole instrument were set to its positions by suspending it from one arm of a balance at H (fig. 3.) the quantity of counterpoise, when in equilibrio, might be applied to determine the pressures to a degree of accuracy much greater than can be obtained by linear measurement.