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79
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A Continuation of p.77.

With respect to the Application of this Theorem to M[[superscript]]r[[/superscript]]. [[underline]]Dolland[[/underline]]'s combined Object-glasses, it is evident, that if the Aberrations of the convex and concave Lenses added together (paying due Regard to the Signs of the Theorem), are made equal to nothing, the two lenses will perfectly correct one another: But as there are two unknown Quantities unlimited in the Equation, namely, the Radius of one Surface of each Glass (for F and Q are given, as well as M and N), there is room for an arbitrary Assumption of one of them, at the Discretion of of the Theorist, or Artist; which being done, there will remain a quadratic Equation, whence there will result two Values of the Radius, which remains unknown, either of which will produce an Aberration equal to that of the other Lens.

[[image - lens, concave mirror, and eye. Labeled with letters A, a, B, b, C-G and S]]

A demonstration of the Speculum in Hadley's Quadrant. Or that the Angle formed by the Speculum at the center, and a perpendicular to the Horizon is equal to the Altitude of the Object taken by the quadrant. By Myself.
     Let AB (in the fig. above) be a plano-speculum, C the Eye of the observer situated in the horizontal line DC & [[perpendicular symbol]] to AB; then if S be a Star, the [[angle symbol]]SDC = Altitude, but since [[angle symbol]]CDS = CDE, by Optics, the Ray SD proceeding ^[[in]] the direction DE, and therefore invisible at C: Wherefore, to have it visible at C, let DF bisect SDC; then [[angle symbol]]SDF, called the angle of incidence = FDC, the [[angle symbol]] of Refraction, consequently, DF is [[perpendicular symbol]] to the Speculum in the position ab.
     Now if the common [[angle symbol]]ADF be taken from the two Right angles ADC, aDF, there will remain FDC = aDA, i.e. half the Altitude of the the Star S is equal to the Angle made by the Speculum and a perpendicular to the Horizon. Wherefore, to measure Altitudes with a Speculum in the center of a quadrant, an Octant will Serve the purpose which must not be divided into 45 but 90 equal parts for degrees.
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A computation of the rise of water against obstacles placed in a running stream, and the fall of the water, on the other side, made thereby.
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On Caleb Smith's Quadrant
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  I have an Old quadrant with the telescope fixed parallel to the limb of Hadley's quadrant, and a solid prismatic glass at the center invented by Caleb Smith. Something similar to which see Philos. Trans. Vol. VI. part I. p. 141. of Martyn's Abridgment. Also Vol. VIII. p. 129.
Stone, in his Appendix to Bion, p. 268. Say's Smith published a plate of his quad[[superscript]]t[[/superscript]].