II. By one Latitude, Rhomb, and Distance, To find the difference of Latitudes.
Let the place given be C in the Latitude of 40 Degrees, that is in the Center of the Quadrant, the second Latitude unknown; The distance upon the Rhomb 21 Leagues 65 parts of a League; the Rhomb N. N. E. the second from the Meridian: Therefore set the Index to the Point, and count 21 Leagues 10 parts, and run your Eye up the Parallel-Line you there meet with, and reckon the Leagues from the Center C to that Line, and you will find it 20 Leagues; and such is the difference of Latitude required.
It is easie to be understood how to lay it down by the Plain Scale; therefore I shall forbear to write any more of that Way.
As the Radius, to the Co-sine of the Rhomb from the Meridian:
So the distance upon the Rhomb, to the difference of the Latitudes.
Extend the Compasses from the Sine of 90, to the Co-sine of the Rhomb 67 deg. 30 m. the same distance will reach from 21-65 Leagues in the Line of Numbers, to the difference of Latitude 20 Leagues. In like manner you must work for all such Propositions, let the Number be greater or less, by either Instrument.
The Travis-Scale is the same manner of Work, as the Artificial Sines, Tangents, and Numbers; For extend the Compasses from 8 Points, to 2 Points, the same distance will reach from 21: 65 in the Line of Numbers, to 20 the difference required.
III. By the Rhomb and both Latitudes, To find the Distance upon the Rhomb.
As suppose the one place given were C the Center of the Quadrant, in the Latitude 40 deg. the second place in the Latitude 41 deg. and the Course the second from the Meridian.
Set the Index to the Rhomb, and account 20 Leagues, which is 41 deg. the second Latitude, and carry your Eye on that Parallel that leads to the Index; and there it will cut the distance upon the Rhomb, which in this Question is 21 Leagues 65 parts.
Extend the Compasses from the Co-sine of the Rhomb from the Meridian, to the Radius or Sine of 90—
The same Extent will reach from 20 Leagues, the difference of Latitude, to 21: 65 in the Line of Numbers, the distance upon the Course required.
IV. By the distance and both Latitudes, To find the Rhomb.
Suppose the Place given was at C, in Latitude 40 deg. and the second Place a Degree or 20 Leagues further Northward, and the distance was 21-65 Leagues upon the Course.
From the Center C reckon 20 Leagues towards D, follow that Parallel, and set the Index, and count the distance until it touch the Parallel, and look in Arch of the Quadrant, and you will find the Rhomb 22 gr. 30 m. or N. N. E. 2 Points from the Meridian.— Or, Extend the Compasses from the distance upon the Rhomb 21: 65, to the distance of Latitudes 20 Leagues; The same Extent will reach from the Radius or Sine of 90, to the Sine-Complement of the Rhomb 67 deg. 30, which was required.
V. By the difference of Meridians, and Latitude of both Places, To find the Rhomb.
As if the Place given was C the Center of the Quadrant, 40 deg. and 20 Leagues was the difference of Latitude Northward, that is 41 deg. and the difference of Longi∣tude 8 Leagues 45 parts of a League.
First, count from C the difference of Latitude 20 Leagues, on that Parallel count 8 Leagues 45 parts; to that put the Index; and in the Arch you will find the Course 22 gr. 30. from the Meridian.
Extend the Compasses from 20 Leagues to 8: 45, the same Extent will reach from 90 to the Tangent of the Rhomb 22 gr. 30 min. as before.