Now therefore, if a certain Rule could be produced, by which the Agreement or Disagreement of the said two Lines might be shew'd, not only to that Extent of places, to which that _Tangent Line_ is already calculated, but also to as many more, as the same may be yet further extended unto, in _infinitum usque_; surely that rule would not only save us the labour of making _Tables_ unto the _Radius_ 10000000; but also the _Helix_ or Spiral Line of the Ships Course would be reduced to a more precise exactness, than ever was pretended by Him: and this most Noble and Useful Science (as He justly calls it) which is the Bond of most disjunct Countries, and the Consociation of Nations farthest remote, would attain its full lustre and perfection. Besides, that the same Rule would also discover a far easier way of making _Logarithmes_, than ever was practised or known; and therefore might serve, when ever there should be occasion, to extend the _Logarithmes_ beyond that number of places, that is already extant. Moreover such a rule would enable men to draw the _Meridian_ line _geometrically_, that is, without _Tables_ or _Scales_: which indeed {217} might also be done, by setting of the _Secants_ of every whole or half degree, if there were not this Inconveniency in it (which is not in my Rule:) That a Line composed of so many small parts, would be subject to many errours, especially in a small compass. The same Rule also will serve, to find the Course and Distance between two Places assigned, as far, as practice shall require it; and that, without any Table of _Meridional_ parts, and yet with as much ease and exactness. And seeing all these things do depend on the solution of this Question, _Whether the Artificial Tangent-line be the true Meridian-line?_ It is therefore, that I undertake, by God's assistance, to resolve the said Question. And to let the world know the readiness and confidence, I have to make good this undertaking, I am willing to lay a _Wager_ against any one or more persons that have a mind to engage, for so much as _another Invention_ of mine (which is of less subtlety, but of far greater benefit to the publick) may be worth to the Inventor.