A chord of a circle is divided in continued proportion, by straight lines inflected to any point in the opposite circumference from the extremities of a parallel tangent, which is limited by another tangent applied at the origin of the chord. Let AB, AC be two tangents applied to a circle, CD a chord drawn parallel to AB, and AE, BE straight lines inflected to a point E in the opposite circumference; then will the chord CD be cut in continued proportion at the points F and G, or CF : CG :: CG : CD, For join BD, BC, and CE. Because the tangent AB is equal to AC (III. 22. cor.), the angle ABC is equal to ACB (I. 10.); but ABC is equal to the angle BCD (I. 22.), and to the angle BDC (III. 21.); whence (VI.1.1.) the triangles BAC and BDC are similar, and AB : BC : : BC : CD, and consequently (V. 6.) BC*=AB.CD. Again, the triangles CBG and CBE are similar, for they have a common angle CBE, and the angle BCG or BCD is equal to BDC or BEC (III. 16.): Wherefore BG: BC: : BC : BE, and BC*=BG.B.E. Hence A.B.CD=BG.BE, and AB : BE : ; BG : CD ; but FG being parallel to AB, AB : BE : ; FG : GE (VI. 2.), and r" - If, from the verter of a triangle, two straight lines be drawn, making equal angles with the sides and cutting the base; the squares of the sides are proportional to the rectangles under the adjacent segments of the base. * . In the triangle ABC, let the straight lines BD and BE make the TE angle ABD equal to CBE ; then AB* : BC*:: DA.AE : EC.CD. For (III. 9. cor.) through the points B, D, and E describe a circle, meeting the sides AB and BC of the triangle in F and G, and { join FG. Because the angles DBF and EBG are equal, they stand (III. 16. cor.) on equal arcs DF and EG, and consequently (III. 18, cor.) FG is parallel to DE. Whence (Vl. 1.) AB : BC :: AF: CG, and therefore (V, 13.) AB’: BC : p AB.AF:BC.CG; but(III. 26.) AB.AF = D.A. AE, T and BC. CG = EC. CD. Wherefore AB* : CD*:: DA.AE : EC.CD. If the triangle ABC be right-angled at C, and the vertical The perpendicular within a circle, is a mean proportional to the segments formed on it by straight lines, drawn from the extremities of the diameter, through any point in the circumference. Let the straight lines AEC and BCG, drawn from the ex tremities of the diameter of a circle The Appendix to the books of Geometry cannot sail, by its novelty and singular beauty, to prove highly interesting. The first part is taken from a scarce tract of Schooten, who was Professor of Mathematics at Leyden, early in the seventeenth century. But the second and most important part is chiefly selected from a most ingenious work of Mascheroni, a celebrated Italian mathematician; which in 1798 was translated into French, under the title of Geometrie du Compas. It will be perceived, however, that I have adapted the arrangement to my own views, and have demonstrated the propositions more strictly in the spirit of the ancient geometry. o NOTES TO TRIGONOMETRY. 1. The French philosophers have, at the instance of Borda, lately proposed and adopted the centesimal division of the quadrant, as easier, more consistent, and better adapted to our scale of arithmetic. On that basis, they have also constructed their ingenious system of measures. The distance of the Pole from the Equator was determined with the most scrupulous accuracy, by a chain of triangles extending from Calais to Barcelona, and since prolonged to the Balearic Isles. Of this quadrantal arc, the ten millionth part, or the tenth part of a second, and equal to 39.371 English inches, constitutes the metre, or unit of linear extension. From the metre again, are derived the several measures of surface and of capacity; and water, at its greatest degree of contraction, furnishes the standard of weights. It would be most desirable, if this elegant and universal system were adopted, at least in books of science. Whether, with all its advantages, it be ever destined to obtain a general currency in the ordinary affairs of life, seems extremely questionable. At all events, its reception must necessarily be very slow and gradual; and, in the meantime, this innovation is productive of much inconvenience, since it not only deranges our habits, but lessens the utility of our delicate instruments and elaborate tables. The fate of the centesimal division may finally depend on the continued inerit of the works framed after that model. 2. The remarks contained in the preliminary scholium, will obviate an objection which may be made against the succeeding demonstrations, that they are not strictly applicable, except when the arcs themselves are each less than a quadrant. But this in fact is the only case absolutely wanted, all the derivative arcs being at once comprehended under the definition of the sine or tangent. To follow out the various combinations, would require a fatiguing multiplicity of diagrams; and |