STEREOGRAPHIC. This word, which is derived from acyclic, "solid," and 7pctcpetr, "to draw," and which therefore ought to be applied to every method of representing a solid in a plane, has never theless a limited technical sense, being applied to that projection of a sphere in which the eye is at a point in the sphere, and the plane of projection is the great circle of which the eye is at the pole, or a plane parallel to it. This mode of projection was known to Hipparchus, and was first described in the work on the plauisphere attributed to Ptolemy.
The stereographic projection has two remarkable (properties. The first is, that all circles are projected either into straight lines or circles. Those which pass through the eye are of course projected into straight lines ; in every other case the projection is the SUCICONTRAllY section of a cone, which has its vertex at the eye, and the circle to be projected for its base ; consequently the projection is a circle. As much of the circle as lies below the plane of projection (the eye being considered as above it) is projected inside the great eirole on which projection is made; and all the rest outside : when this projection is employed in maps, it is usual to place all the part of the globe to bo projected below the plane of projection.
The second property is, that the angle made by two circles which meet on the globe, is equal to the angle made, at the point of meeting, by the two circles which are the projections of those circles, the angle made by two intersecting circles being always that made by their tangents. This property is easily proved as follows : Draw through the point'of intersection of the two circles (A and B) which are to be projected, two other circles (A' and u'), which have the same tangents, and pass through the eye. Then the tangents of A' and B' at the eye make the same angle as those nt the other point of intersection* that is, as the tangents of A and B at the point to be projected. But these tangents of A' and B' at the eye are parallel to the projections of the tangents of A and n at the point to be projected : whence the projections of these tangents of A and B make the same angle as the tangents themselves.
The first property was known to Hipparchus and Ptolemy : the history of the second is rather curious. The first writer who seems to have looked attentively for a discoverer was Delambre (` Wm. Inst.', vol. v., p. 393), who could not find it in Clavius, Stoffler, nor in any of the writers of the middle ages, who have treated pretty voluminously on the astrolabe, which word, as used by them, merely meant a stereo graphic projection. That it was mentioned (without demonstration) in the French Mathematical Dictionary of Sav6rien (1753), in an article which was copied word for word into the Encyclop6die; was all that Delambre could then say of its origin. He afterwards, in
writing his History of Astronomy in the Middle Ages,' found the proposition demonstrated in the Compleat System of Astronomy,' by Charles Leadbetter, London, 1728 ; but, judging from the rest of the work, he presumes that Leadbetter could not have been the discoverer. No claim was, however, at the time put in for any one else, and Sav6rien's article, which appeared in the EncyclopMie; first called general attention to the property, and this can he traced to Lead better's work nearly. For we find that Sav6rien translated his article, word for word, from the second edition (1743) of Stone's Mathematical Dictionary.' Stone was a contemporary of Leadbetter, and several times refers to his writings.
On consulting the third edition of Dr. Harris's 'Lexicon Technicum ' (1716), and feeling sure, with regard to that work, that such a proposition as the one called Leadbetter's would be stated, if it were then known, we turned to the article' Spherick Geometry,' and there we found it, with a demonstration, enunciated as follows :—" All Angles made by Circles on the Superfieies of the Sphere are equal to those made by their Representatives on the Plane of the Projection." The claim of Leadbetter is therefore overthrown. In the preface, I farris says that under (among others) Spherical Geometry' will be found entire treatises, which, if he mistakes not, arc as short and plain as any extant. If this proposition had been new, he would probably have noted it here, particularly if it had been his own. We find how ever, finally, that the property has been shown (' Encyc. Brit.', 'Pro jection ') to have been demonstrated by Halley in No. 219 of the ' Philosophical Transactions,' and is attributed by him to De Moivre or I look.
The consequence of this theorem is, that any small portion of the sphere is projected into a figure very nearly similar to itself, so that any not very large portion of the earth preserves its figure with tolerable accuracy in the map. 4fence some writers have said that there is no distortion in the stereographic projection, which is not absolutely true, though nearly so of countries which bear no greater proportion to the whole earth than most of them.
The mode of laying down the stereographic projection is briefly stated, and a diagram given, in the article MAP; it will be found at greater length in the memoir of Delambre above cited, or in any good work on the construction of maps.