What is the great Circle navigation ?

• Great Circle Navigation is the chiefest of all the 3 kinds of sayling, in whom all the other are contained, and by them this kind of sayling is perfomed, continuing a Corse by the shortest distance between places, not limited to any one Corse, whether horizontal or paradoxal, but by those Corses are Ordered to the full perfection of this rare practice, whole benefits in long Voyages are to great purpose, ordering and disposing all horizontal traverses to a perfect conclusion:
• for there are many changes of horizontal and paradoxal Corses in the execution of this practise, so that upon the shifting of a wind, when that it may seem that you are forced to an inconvenient corse by the skill of great Circle sayling, that corse shall be found the shortest and only proper motion to perform your voyage.
• And also when with favourable winds the pilote shall shape a Corse by his Chart or Compasse paradoxall, as the best means to attaine his Port, he shall by this kinde of sayling finde a better and shorter Corse, and by sufficient demonstration prove the same, so that without this knowledge I see not how Corses may be ordered to their best advantage:
• therefore with by it the perfection of sayling is largely understood, and the error likewisemost substantially controlled, it may of right challenge the chiefest place among the practises gubernautik. The particularities whereof if I should by an orderly method labour to express, it would be a discourse over-large for this place, nor I think troublesome if the premises be not well understood: therefore I will now over-pass it, until a time more convenient, and of better pleasure.

Of Paradoxal Navigation.

• Paradoxal Navigaton demonstrateth the true motion of the ship upon any Corse assigned, in his true nature by longitude, latitude, and distance, giving the full limit of determination of the same, by which motion, lines are described neither circular nor strait, but concurred or winding lines, and are therefore called paradoxal, because it is beyond opinion that such lines should be described by plain horizontal motion:
• for the full perfection of which practise I purpose (if God permit) to publish a paradoxal Chart, with all convenient speed and so will discover by the same at large all the practices of paradoxal and great circle navigation, for upon the paradoxal Chart, it will best serve the Seamans purpose. being an instrument portable, of easie stowage and small practise, performing the practises of Navigation as largely and as beneficially as the Globe in all respects: and all these practices of sayling before mentioned, may in a general name be aptly called Navigaton geometrical, because it wholly consisteth of Geometrical demonstrative conclusions.
• But there is another knowledge of Navigation, which so far excelleth all that is before spoken, or that hach hitherto been vulgarly practised, as the substance his shadow or as the light surpasseth the thick obscured darkness: and this sweet skill of sayling may well be called Navigation arithmetical, because it wholly consisteth of calculations, comprehended within the limit of numbers, distingushing corses not only upon the points of the compass, but upon every degree of the horizon, and giveth the distance of any Travers for the particular elevation of minutes, yea, and less parts assure your self: it giveth longitudes and latitudes to the minute, second, and third:
• in so great certainty, as that by other means the like cannot be performed: it teacheth the nature of angles and triangles, as well spherical as plain superficial and soled commenformations, the effort of lines Grais*, circular, & paradoxal, the quantities and proportions of parallels, the nature of Horizons, with every particular distinction of any alteration whatsoever, that may in Navigation be required to a most wonderful precise certainty:
• for there can nothing be required, that by this heavenly harmony of numbers shall not be most copiously manifested to the Seamans admiration & great content: the orderly practice thereof, to the best of my poor capacity, I propose to make known, if I may perceive my pains already taken, to be received in good part, which I distrust not but all honest-minded Seamen & Pilots of reputation will gratefully embrace, only in regard of my friendly good will towards them, for it is not in respect of my pains, but of my love, that I would receive favorable curtesie.