CHAP. XIV. How by the Scale of Tangents to make a Part of the Globe in Plano, where∣by you may trace out the Latitudes to every Degree of Longitude; or eve∣ry 5 or 10 Degrees, as neer as you will desire, without Calculation.

BY the Line of Tangents on the side of your Mathematical Scale, you may make the following Projection, which was made by Mr. Philips in his Geo∣metrical Seaman, pag. 5. by Tables and Geometry: But here you may save that labour, if you have a Scale with a Line of Tangents on it.

First, Consider of what length your Tangent or Side of your Quadrant must be, and accordingly set off your Radius from A towards D, as I have done, by taking off 77 deg. of the Tangent-line of my Scale, and set it from A the Pole to D, for 13 deg,* on the North side of the Aequator, or 13 deg. of North Latitude, which is the Com∣plementPage  194 of 77 to 90 deg. Then make the other side of the same length, and draw the Quadrant ADE, the Radius is always a Tangent of 45 deg. Then with your Com∣passes take off the Line of Tangents the several Degrees, and draw the Arches or Parallels of Latitude, as you see I have done in the Figure. Thirdly, divide the Limb of the Arch DE into 90 deg. and through every 5 or 10 deg. draw Lines of Longitude, or Meridian-lines. The Arches of Latitude must be numbred as in the Fi∣gure; but the Lines of Longitude you may number from either side, as occasion re∣quires.

You may, if you will, when occasion requires, divide a Circle into four Qua∣drants, and draw the Lines of Longitude from the Center; and you may make this as large or as little as you will, by the Tables of Natural Tangents in the Second Book, as you have been there shewed how to lengthen or shorten your Radius: You may number the whole Circle of Longitude into 360 deg.

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The Blank Quadrant, being thus made, will serve for many Examples; especially if you make it upon a Slat Stone, that you may wipe the Arch, that is lightly drawn by a Slat Pen betwixt any two Places, off at pleasure.

You may set down therein the two Places you are to sail between, according to their Latitudes and Longitudes; and then only by your Ruler draw a streight Line from the one Place to the other, which will represent the Great Circle which passeth between the two Places, and will cross those Degrees of Longitude and Latitude, which you must sail by exactly. You may do it by the Difference of Longitude only, if you will, as shall be shewed in this Example, for proof thereof.

Of a Voyage from Lundy, in Latitude 51 deg. 22 min. and Barbadoes, in Longitude 332 deg. 27 min. Difference 52 deg. 55 min. and Latitude 13 deg. 10 min. To find by what Longitude and Latitudes the Arch of a Great Circle drawn between those two Places doth pass.

First, Let the Line AD represent the Meridian of the Island of Lundy,* marked out by L for its Latitude 51 deg. 22 min. and the Longitude thereof 25 deg. 52 min. at D, which is set down according to its Longitude and Latitude. Then from D in the Limb or Arch of the Quadrant, count the Difference of Longitude 52 deg. 55 min. and this is the Meridian of the Island of Barbadoes, on which you must mark out the Latitude 13 deg. 10 min. at B; lay a Ruler from the first Latitude L to the second at B, and draw the streight Line LB, which representeth the Arch of a Great Circle between the two Places; and if you guide your Eye along in this Line, you may rea∣dily and truly perceive by what Longitudes and Latitudes you should sail: For where this Line crosseth the Arches of Latitude and the Lines of Longitude, that shews the true Longitude and Latitude of the Arch, accord∣ing to your desire.

Differ. of Longit. substract	Differ of Longit. from Obliqu.	Latitude	Differ of Longit. added.
D. M.	D. M.	D. M.	D. M.
52 55	27 32	51 22	00 00
2 28	30 00	50 41	02 28
5 00	35 00	49 07	07 28
5 00	40 00	47 13	12 28
5 00	45 00	44 56	17 28
5 00	50 00	42 12	22 28
5 00	55 00	38 58	27 28
5 00	60 00	35 12	32 28
5 00	15 00	30 48	37 28
5 00	70 00	2• 46	42 28
5 00	75 00	10 30	47 28
5 27	80 27	13 10	52 55
52 55		Barbad.

Now the truth hereof will more evidently appear, if you compare the Latitudes and Longitudes which this Line intersecteth, with this Table, as before, Cal∣culated by me for every •deg. of Longitude. You may see by the Figure, that the Line BL in the Points a, b, c, d, e, F, g, h, i, k, L, doth cross the Parallel of Latitude, as you see in the fourth Column of this Table, at the same number of Degrees from the first Me∣ridian, as you see in the fifth Calu•n. The first Column is the Number of Degrees of Longitude, the second is the Difference sub∣stracted, the third is the Degrees of Longi∣tude from the Meridian of the greatest Ob∣liquity, the fourth and fifth are, as before, Latitude and Longitude added, for the Dif∣ference of Longitude from Lundy. For Example: I would know what Degree of Latitude 47 deg. 28 min. of Longitude from the first Meridian doth cross; and I see by the Figure it is at •0 deg. of Longitude, which the Table sheweth is 20 deg.•0 m. of Latitude, and so of the rest, 75 deg. from Meridian of Obliquity, and Longitude 338 deg. 24 min.

And so in like manner you may lay down upon the former Quadrant any two Places, howsoever scituated, by their Longitude and Latitude, of Difference of Longi∣tude, in the manner as you have been shewed in the last Example.