# The First Book of the Cross-Staff

## CHAPT. III. The Use of Tangent Lines to taking Angles.

1. To find an Angle by the Tangent on the Staff

Let the middle sight be always set to the middle of the Cross, noted with the 20 and 30, and then the Cross drawn nearer to the eye, until the marks may be seen close within the sights.For so if the eye at A (that end of the Staff which is noted with 90 and 180) beholding the mark K and N between the first two sights, C and B, or the marks K and P between the two outward sights, the Cross being drawn down unto H, shall stand at 30 and 60, in the Tangent on the Staff: it sheweth the Angle KAN is 30 gr. the Angle KAR is 60 gr. the one double to the other; which is the reason for the double numbers on this Line on the Staff: and this way will serve for any Angle from 20 gr toward 90 gr. or from 40 gr. toward 180 gr. But if the Angle be less than 20 gr. we must than make use of the Tangent upon the Cross.

2. To find an Angle by the Tangent of 20 upon the Cross.

Set 20 unto 20, that is, the middle sight to the midts of the Cross at the end of the Staff, noted with 20, so the eye at A, beholding the marks L and N, close between the first two sights, C and B, shall see them in an Angle of 20 gr.

If the marks shall be nearer together, as are M and N, then draw in the Cross from C unto F; so the quantity of the Angle shall be found on the Cross in the Tangent of 20 gr. at the end of the Staff: and this will serve for any Angle from 20 gr. towards 35 gr.

3. To find an Angle by the Tangent of 30 upon the Cross.

This Tangent of 30 is here put the rather, that the end of the Staff resting at the eye, the hand may more easily remove the Cross; for it supposeth the Radius to be no longer than AH, which is from the eye ath the edn of the Staff unto 30 gr. about 22 inches and 7 parts. Wherefore here ste the middle sight uto 30 gr. on the Staff, and than either draw the Cross in or out, until the marks be seen between the two first sights; so the quantity of the Angle will be found in the Tangent of 30, which is here represented by the Line GH; and this will serve for any Angle from 0 gr. toward 48 gr.

4. To observe the altitude of the Sun backward.

Here it is fit to have an horizontal sight set to the beginning of the Staff, and than may you turn your back toward the Sun, and your Cross toward the eye. If the Altitude be under 45 gr. set the middle sight to 30 on the Staff, and look the middle sight through the Horizontal unto the Horizon, moving the Croff upward and downward, until the upper sight do shadow the uper half of the horizontal sight: so the Altitude will be found in the Tangent of 30.

If the Altitude shall be more than 45 gr. set the middle sight unto the midst of the Cross, and by the inward edge of the lower sight through the Horizontal to the Horizon, moving the middle sight in or out, untill the upper sight do shadow the upper half of the Horizontal sight: so the Altitude will be found in the degrees on the Staff between 40 and 180.

5. To set the Staff to any Angle given.

This is the converse of the former Proposition: For if the middle sight toi be set to his place and degree, the eye looking close by the sights as before, cannot but see his object in the Angle given.

6. To observe the Altitude of the Sun another way.

Set the middle sight to the middle of the Cross, and hold the Horizontal sight downward so as the Cross may be parallel to the Horizon, then is the Staff vertical; and if the outward sights of the Cross do shadow the Horizontal sight: the Compliment of the Altitude will be found on the Tangent of the Staff.

7. To observe an Altitude by Thread and Plummet.

Let the middle sight be set to the midst of the Cross, and to that end of the Staff which is noted with 90 and 180; than having a Thread and a Plummet at the beginning of the Cross, and turning the Cross upward, and the Staff toward the Sun, the Thread will fall on the Compliment of the Altitude above the Horizon. And this may be applied to other purposes.

8. To apply the Lines of Inches to the taking of Angles.

If the Angles to be observed between the first two sights, ther will be such proportions between the poarts of the Staff and the parts of the Cross, as between the Tadius and the Tangent of the Angle.

As if the parts intercepted on the Staff were 20 inches, the parts on the Cross 9 inches. Then by proportion as 20 unto 9, so 10000 unto 45000 the Tangent 24 gr. 14m.

But if the Angle shall be observed between the two outward sights, the parts being 20 and 9 as before, the Angle will be 48 gr. 28 m. double uto the former.

In all this there is a regard to be had to the Parallax of the eye, and his height above the Horizon in observations at Sea; to the semidiameter of the Sun, his parallax and refraction, as in the use of other staves. And so this will be as much, or more than that which hath been heretofore performed by the Cross-Staff.

Zurück zum Inhaltsverzeichnis Nächstes Kapitel