VOL. XV.J 1PHIL080PHICAL TRANSACTIONS. 229 



hose answering to the radius so divided, are, — , taking 5 in arithmetical pro- 

 gression. And then, enlarging the base, as in fig. 8, or the tangent as in fig. 9. 

 in the proportion of the tangent to the sine; as*: — ::c:r::— : — = 



-J— I . Hence, by division, we have this series, s 4--i--j- - — \. .L. ^ ^ gj^.. 



That is, putting r = 1, the series j + s'' + 5' + *' + s^, &c: which multiplying 

 the respective members by 4-s, -j-j, -^s, i-s, -^s, &c. becomes x*'^ + ^s* + 

 is" + -ys^ + -tV*'^ &c. which is the aggregate of tangents, to the arch whose 

 right sine is s. And this method may be a pattern for the like process in other 

 cases of similar nature. 



^ Letter from Mr. S. George 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. N° J 76, p. 1202. 



This infirmity did not show itself on the subject of this letter, viz. Ann 

 Jackson, till she was about 3 years old, after which the mother concealed her 

 out of shame, and brought her up privately; but she soon dying, and the father 

 becoming exceedingly poor, the child was left as a charge upon the parish. She 

 is now between 13 and 14 years of age, yet can scarcely go, and is so little in 

 stature, that I have seen children of 5 years old taller ; she is very silly, speaks 

 but little, and that not plainly, but hastily, and with difficulty ; her voice is low 

 and rough : her complexion and face well enough, except her eyes, which look 

 very dead, and seem to have a film or horn growing over them, so that she can 

 hardly now perceive the difference of colours. 



The horns abound chiefly about the joints and flexures, and not on the 

 brawny fleshy parts of her body ; they are fastened to the skin like warts, and 

 about the roots resemble them much in substance, though toward the extre- 

 mities they grow more hard and horny ; at the ends of each finger and toe grows 

 one as long as the finger or toe, not straight forward, but rising a little between 

 the nail and the flesh (for near the roots of these excrescences is something 

 like a nail) and bending again like a turkey's claw, which also it much resembles 

 in colour ; on the other joints of her fingers and toes are smaller ones, which 

 sometimes fall off, others growing in their places. The whole skin of her feet, 

 legs, and arms, is very hard and callous, and grows always more and more so ; 

 on her knees and elbows, and round about the joints, are many horns ; two 

 more remarkable at the point of each elbow, which twist like rams' horns ; that 



