VOL. VII.] PHILOSOPHICAL TRANSACTIONS. 41 



itself, or EA, or parallel thereto; as will easily be found by the data. And these 

 various cases may be explained by the equations for the circle. 



Thus, let the diameter of a semicircle be EB, (Fig. 3), and let D be a given 

 point, from which may fall the perpendicular AT) = v. Let B A = y, BE = ^*; 

 then the equation will be hy — yy =. vv\ and drawing the tangent DC, it will 



be AC, or a = ,_ n • Now if b be greater than 2?/, the tangent is to be 

 drawn towards B ; if less, towards E; if equal to it, it will be parallel to EB ; 

 as was said in N° 1,2, 4. 



Let there be any semicircle inverted, as NDD, (fig. 4), the points of whose 

 periphery are to be referred to the right line BE, parallel and equal to the dia- 

 meter. Making NB =2 d, and all things else as above, gives the equation by — 



o 71 71 __„ Q,dv 



yy =dd-\-vv — 2 f/i; : therefore A C or (2 = — . — . Now since v here is 



supposed to be always less than «?; if ^ be greater than 2y, then the tangent 

 must be drawn towards E ; if equal, it will be parallel to BE ; if less, changing 

 all the signs, the tangent must be drawn towards B; as by N°4, 5, and 3. But 

 there could be no tangent drawn, or at least, EB would be the tangent, if NB 

 had been taken equal to the semidiameter, or 2'd = Z?, as by N** 5. 



Let there be another semicircle, whose diameter NB, (fig. 5), is perpendi- 

 cular to EB, and to which its points are supposed to be referred. Let NB be 

 called bj and all things standing as before ; then the equation will he yy = bv 



— vv, and hence a = —^ Now if Z) be greater than 2 v, the tangent must 



be drawn towards B; if less, towards E; but if equal, then DA will be the tan- 

 gent ; as by N° 1 , 4, and 5 . And these are all the various cases that the con- 

 sideration of equations can afford. 



But how the limits of equations are derived from this doctrine of tangents, I 

 do not explain, being a thing evident; and the application to the maxima or 

 minima, which are determined both at once by the parallel tangent: concerning 

 which, and other matters I have written to you, and have also treated some- 

 what on them in my Miscellanies ; where I have also shown how to find the 

 points of contrary flexure from the tangents. The same method may be applied 

 to other things also, too long to be shown in this letter. 



^n Account of some Books. N° 90, p. 5147. 



I. A Discourse concerning the Origin and Properties of Wind, &c. By 

 R. Bohun, Fellow of N. Col. in Oxon. 1671, in 8vo. 



After deducing several inferences, from such relations as the author has ob- 

 tained from books and travellers, he concludes, 



VOL. II. G 



