An Explanation of the Figures of Several Antiquities, Communicated by a Member of the Royal Society

58. For, those answering to the Radius so divided, are $\frac{SR}{S}$; (taking $S$ in Arithmetical progression.) 59. And then, inlarging the Base (as in Fig. 8.) or the Tangent (as in Fig. 9.) in the proportion of the Tangent to the Sine. $$S \cdot \frac{SR}{S} (\because \Sigma R) :: \frac{SR}{S} \cdot \frac{SR_2}{S_2} = \frac{SR_2}{R_2 - S_2}.$$ 60. We have (by division) this Series, $$R^2 S^2 \left( \frac{SR^2}{S}, \frac{S_3}{R_2}, \frac{S_5}{R_4}, \ldots \right)$$ $$S + \frac{S_3}{R_2} + \frac{S_5}{R_4} + \frac{S_7}{R_6} + \frac{S_9}{R_8} \&c.$$ 61. That is (putting $R=1$) $$S + S^3 + S^5 + S^7 + S^9, \&c.$$ 62. Which (multiplying the respective members by $\frac{1}{2} S$, $\frac{1}{4} S$, $\frac{1}{8} S$, $\frac{1}{16} S$, &c.) becomes $$\frac{1}{2} S^2 + \frac{1}{4} S^4 + \frac{1}{8} S^6 + \frac{1}{16} S^8 + \frac{1}{32} S^{16}, \&c.$$ Which is the Aggregate of Tangents to the Arch whose right Sine is $S$. 63. And this method may be a pattern for the like process in other cases of like nature. An Explanation of the Figures of Several Antiquities, communicated by a Member of the Royal Society. FIG. 10, 11, 12, 13, 14, 15. Res Turpiculae, or Priapi, worn by Roman Children against Fascination. 16. An Egyptian brass Serapis, or Teraphim. 17. A brass Stilus Scriptorius. K k k 2 18, 19. Old 18, 19. Old Roman Keys of brass; one being a Ring to wear on a Woman's finger. 20. An Iron Die, or Alea. 21. A flat Iron Die, or Talus; upon the narrow sides are 6, 5, 2, and 1. 22. A Roman Iron Ring. 23. An old Roman brass Ring, marked XXXV, for a slave to wear. 24. A brass Roman Ear-Ring. 25. A brass Lunula, or Meniscus. 26. A brass Fibula. 27, 28, 29, 30, 31. Antient Pastes, or Opaque-Enamels of divers colours, for Pavements; taken up at Baiae. The Figures 32, 33, 34. represent the Cochineel Fly, as seen on its belly by the help of the Microscope, and by the naked Eye, and as seen on its back thro' a Microscope; the Draughts of which were communicated by Dr Tyson, Fellow of the Royal Society. A Letter from Mr. St Georg Ash, Sec. of the Dublin Society, to one of the Secretaries of the Royal Society; concerning a Girl in Ireland, who has several Horns growing on her Body. Trin. Coll. the 10th of Octo. 1685. Hon. Sir, The Account I here send of the Horny Girl is, much more imperfect than I hoped it would have been, both because its parents or friends, who might give some information of the beginning and occasion of the growing out of these Horns, are not to be found, and that the owner of this Monster would not be persuaded to let us take the figure thereof, which we designed to present you.