Corol. 5. If the specula be supposed to be at the centre of an'infinite sphere, objects in the circumfer ence of a great circle, to which their common section is perpendicular, will appear removed by the two re flexions, through an arch of that circle, equal to twice the inclination of the specula, as before said. But objects at a distance from that circle will appear re moved through the similar arch of a parallel; there fore the change of their apparent place will be mea sured by an arch of a great circle, whose chord is to the chord of the arch equal to double the inclination of the specula, as the cosines of their respective dis tances from that circle are to the radius; and if these distances are very small, the difference between the apparent translation of any one of these objects, and the translation of those which are in the circumfer ence of the great circle aforesaid, will be to an arch equal to the versed sine of the distance of this object from that circle, nearly as double the sine of the angle of inclination of the specula, is to the cosine of the same.
For let OBC, Fig. 4. in the annexed figure, repre sent an infinite sphere, at whose centre R are placed the two specula inclined to one another in any given angle, and let their common section coincide with the diameter ORC. Let BAN be the circumference of a great circle, to the plane of which the common sec tion of the specula ORC is perpendicular, and BR its radius; let b a n be the circumference of a circle pa rallel to BAN, and at the distance from it B b; draw b D the sine, and b r the cosine of the arch 11 b; BD is the versed sine of the same.
Let A be a point of an object placed in the circum ference of the great circle BAN, and N the point in which its image is formed by the two successive re flexions as before described; and let a be a point of another object placed any where in the circumference of the parallel b a n, and n its image; and let a b n be an arch of a great circle passing through the points a and n. The point a is at the same distance from the great circle BAN, as the point b, i. e. at the dis tance 13 b. Draw AR, AN, RN, a r, a n, r n, a R, and n R.
By the fourth corollary the figures ARN. and a r n are similar, and consequently the line AN is to the line a n as AR or BR is to a r or b r, i. e. as the ra dius is to the cosine of the distance B b. But AN is the chord of the arch AHN of the great circle BAN, equal to the translation of the point A, or double the inclination of the specula, and a n is the chord of the arch a it n of a great circle measuring the angle a R n, by which the point a appears removed by the two re flexions, to an eye placed in the centre R. Therefore
the translation, or apparent change of place, of the point a is measured by an•arch of a great circle, whose chord is to the chord of the arch AIIN, (equal to dou ble the inclination of the specula,) as the cosine of its distance from the great circle BAN is to the radius.
From any point C of the circumference OBC draw the chords CN and C in, to the same side of the point C, and equal to the chords AN and a n respectively, draw the radius RM, and from R and m draw RQ and P, both perpendicular to CM, and cutting it in Q and P. RQ is the cosine, and CM double the sine of half the angle MRG, or ARN, or of the angle of incli nation of the specula. The little arch Al m will re present the difference of the apparent translations of the objects in A and a; and if it be very small, may be looked on as a straight line, and the little mixed triangle M m P as a rectilinear one, which will be similar to RMQ, because RN1 is perpendicular to M in and RQ to CM, and the angles at Q and P right angles.
The line CP may be taken as equal to Cm and MP as the difference of the lines CM and C in. There fore the little arch M 772 is to the line MP nearly as RM to RQ; hut CM (i. e. AN) was to C m e. a 71) as BR to b r, and the difference MP, of CM and C m, to the difference BD of BR and b r, as CM to BR. Therefore M nt, the difference of the apparent trans lations, is to BD, the versed sine of the distance B b, or to an arch equal to it, in the compound ratio of RM the radius to 11Q, the cosine of the angle of incli nation of the specula, and CM double the sine of the same to BR the radius, i. e. as CM to RQ, The observation may be corrected by one easy ope ration in trigonometry, as will appear fro.n the first part of this corollary, viz. by taking the half of the angle observed and then finding another angle whose sine is to the sine of that half, as the cosine of the dis tance B b is to the radius: this angle doubled, will be the true distance of the object. But as this operation, though easy, will require the use of figures, the me thod of approximation is better, because by that, the observer retaining in his memory the proportion of the sines of a few particular arches to the radius,may easily estimate the correction without figures, when the angle is not great, and by a line of artificial num bers and sines, may always determine it with greater exactness than will ever be necessary.