The First Book of the CrossStaffCHAPT. VIII. The use of the Line of Artificial Tangents.This Line of Tangents hath like use, but commonly joyned with the Line of Sines: the manner of working by it, may appear by this example:
This Proposition, and such others upon two Lines, may be wrought two ways. For extend the Compasses from the Tangent of 38 gr. 30 m. the Tangent of 23 gr. 30 m. the same extent shall give the distance from the Sine 90 gr. to the Sine of 33 gr. 8 m. Or else extend them from 38 gr. 30 m. in the Tangents unto 90 gr. in the Line of Sines; the same extent from the Tangent of 23 gr. 30 m. shall reach to the Sine of 33 gr. 8 m. which is the fourth proportional Sine required. And this Corss work in many cases is the better, in regard the Tangents which should pass on from 40 gr. to 50 gr. and so forward, do turn back at 45 gr. These two Lines of Sines and Tangents, may serve for the resolution of all Sherical Triangles, according to those Canons which I have set down in the use of the Sector. Only two cases the 19 and 20 will be more easily resolved by that which followeth in the last Chapter of this book. Or if at any time one meet with a Secant, Let him account the Sine of 80 gr. for a Secant of 10 gr. and the Sine of 70 gr. for a Secant of 20 gr. and so take the Sine of the Complement instead of the Secant.
Extend the Compasses from the Radius that is the Sine of 90 gr. to the Sine of 38 gr. 30 m. the same extent will give the distance from the Sine of 23 gr. 30 m. both the Sine of 14 gr. 22 m. to the Sine of 39 gr. 50 m. But in this case, the Sine of 39 gr. 50 m. is the fourth required. For the first number being less than the second, that is, the Radius less than the Secant, the Sine of 23 gr. 30 m. which is the third, must also be less than the fourth. If the fourth proportional number shall at any time fall out of the Line, by reason of the minutes that are wanting in the first degree, it imay be supplied by resolving the third number given into minutes, and then working by the Line of Numbers.
Extend the Compasses from the Sine of 90 gr. unto the Sine of 10 gr. the same extent will reach from the Sine or Tangent of 5 gr. beyond the end of the Staff. Wherefore I resolve these 5 gr. into 300 m. and find the former extent to reach in the Line of Numbers from 300 m. unto 52 m. and such is the fourth proportional required. If the extent from the Sine of 90 gr. unto the Sine of 10 gr. be too large for the Compasses, we may use the Sine of 5 gr. 44 m. instead of the Sine of 90 gr. And so extending the Compasses from the Sine of 5 gr. 44 m. unto the Sine of 10 gr. we shall find the same extent to reach in the Line of Numbers fromm 300 unto 52 as before. And by the same reason we may use the Tangent of 5 gr. 43 m. instead of the Tangent of 45 gr. as I further shew in the next Capter. 

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