340 Mr DAVIES on the Equations of Loci 



It requires but a moment's reflection to shew that these two tables re- 

 present curves perfectly equal in all respects, but reversed in position. If 

 we cause the sphere, upon which cot a = i is traced, to turn upon PP 

 half a revolution, and then to turn upon the diameter perpendicular to the 

 plane of the paper half a revolution also, the whole system will become 

 identical with that represented in the figure to cot a = + i, provided i have 

 the same numerical value in both cases. 



To state consecutively the course of the curve, then, let us confine our 

 attention to the former figure (25.) and its corresponding analytical table. 

 Departing from in the negative direction (that is, to the left), the curve 

 winds round the pole, an infinite number of times ; after which it passes 

 through the pole, making with the first meridian POP' the same angle a. 

 It then winds round the pole P, in the positive direction (that is, to the 

 right), and departing farther at each turn, cuts the equator, in a point O, 

 diametrically opposite to O. At this point its direction of revolution be- 

 comes negative again, and it proceeds into the other hemisphere, passes 

 through P' after an infinite number of circumvolutions, and returns by the 

 positive direction round the pole through the extent of the same lower 

 hemisphere till it arrives at the origin O, where it coalesces with the branch 

 along which we set out. The course of the curve is therefore completely 

 assigned. 



We are thus enabled to answer satisfactorily a curious question that was 

 much agitated by the earlier writers upon nautical subjects; viz. "Whether 

 a ship, by sailing on the same rhumb, would ever return to the place 

 from which it set out?" Dr WILSON, in his History of the Rise and 

 Progress of Navigation*, tells us, that Mr JOHN BASS AT, in a post- 

 humous work, published about 1630, established this proposition in the af- 

 firmative. Though I have long sought for that little work (" Pathway to 

 Perfect Sailing"), I have sought in vain ; and am therefore unable to give 

 any account of the manner in which BASSAT has performed this under- 



* Prefixed to Robertson's Navigation. The remark is in the foot-note at p. XT, xvi, 



of Wales's edition (or fourth) of that work. 

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