At Fig. 4, one of the hollow glass parallelopipeds on an enlarged scale is represented.
MATHEMATICS.
Art. XX. An improved Method of obtaining the Formulæ for the Sines and Cosines of the Sum and Difference of two Arcs.
Art. XX. An improved Method of obtaining the Formulæ for the Sines and Cosines of the Sum and Difference of two Arcs, by Professor Strong, of Hamilton College.
In the circle ABCD let AB and BC denote any two arcs contiguous to each other. Draw their limiting diameters Aa, Cc; their sines Bx, By; and join x, y. Then will xy = sine of (AB + BC): for if upon OB as a diameter we describe a circle, it will manifestly pass through the points x and y, (since the angles OxB, OyB are right, see Euc. 31. 3.) therefore OxBy is a quadrilateral inscribed in a circle described on OB as a diameter, and the angle yOx at the circumference stands upon an arc whose chord is xy. Again, if from a we draw ad perpendicular to Cc, it will be the sine of the arc ac (= AB + BC). If now we describe a circle on aO as diameter, it will pass through d, (see Euc. 31. 3.) therefore ad is the chord of an arc on which the angle aOc stands in the circle described on aO. But in equal circles the chords of arcs on which equal angles at the centres or circumferences stand are equal; (see Euc. 26. and 29. 3.) hence xy = ad = sin(AB + BC). Now sine OxBy is a quadrilateral inscribed in the circle described on OB as diameter, we shall have (Euc. D. 6.) OB · xy = Bx · Oy + By · Ox = sinAB · cosCB + sinCB · cosAB. If OB be denoted by r, we shall have xy, or sin(AB + BC) =