First, draw a Circle ABDC, which cross at right Angles with the Diameters AD, CB; then continue out AD to G, and upon the Point B, raise BF perpendicular to CB. Now draw the Chord AB, and divide the Quadrant AB into 9 equal Tarts, setting the Figures 10, 20, 30, &c. to 90 to them; each of which 9 Parts again subdivide into 10 more equal Parts, and then the Quadrant will be divided into 90 Degrees. Now setting one Foot of your Compasses in the Point A, transfer the said Divisions to the Chord Line AB, and set thereto the Figures 10, 20, 30, &c. and the Line of Chords AB, will be divided, and then may be put upon your Scale, represented in *Fig.* 6. Now to project the Sines, divide the Arc BD into 90 Degrees, as before you did AB; from every of which Degrees, let fall Perpendiculars on the Semidiameter EB; which Perpendiculars will divide EB into a Line of Sines, to which you must set 10, 20, 30, &c. beginning from the Center, and then you may transfer the Line of Sines to your Scale.

Again, to project the Line of Tangents; from the Center E, and thro’ every Division of the Arc BD, draw right Lines cutting BF, which will divide it into a Line of Tangents, setting thereto the Numbers 10, 20, 30, &c. which you must transfer to your Scale.

To project the Line of Secants, transfer the Distances E 10, E 20, E 30, &c. that is, the Distance from E to 10, 20, 30, &c. on the Tangent Line, upon the Line EG, and setting thereto the Numbers 10, 20, 30, &c. the Line EG will be divided into a Line of Secants, which must be transferred on the Scale.

To project the Semi-tangents; draw Lines from the Point C, thro’ every Degree of the Quadrant AB, and they will divide the Diameter AE into a Line of Semi-tangents: but because the Semi-tangents, or Plane-Scales of a Foot in Length, run to 160 Degrees, continue out the Line AE, and draw Lines from the Point C, thro’ the Degrees of the Quadrant CA, cutting the said continued Portion of AE, and you will have a Line of Half-tangents to 160 Degrees, or further, if you please.

*Note*, The Semi-tangent of any Arc, is but the Tangent of half that Arc, as will easily appear from it’s manner of Projection, and Prop. 20. Lib. 3. Eucl. where it is proved, that an Angle at the Center, is double to one at the Circumference.

Moreover, to draw the Rhumb-Line; from every 8th part of the Quadrant AC, setting one Foot of your Compasses in A, describe Arcs cutting the Chord AC, which will divide AC into a Line of whole Rhumbs, and in the same Manner may the Subdivisions of half and quarter Rhumbs be made.

Lastly, to project the Line of Longitude; draw the Line HD, equal and parallel to the Radius CE, which divide into 60 equal Parts (because 60 Miles make a Degree of Longitude under the Equator), every 10 of which Number set Figures to. Now from every of those Parts, let fall Perpendiculars to CE, cutting the Arc CD; and having drawn the Chord CD, with one Foot of your Compasses in D, transfer the Distances from D, to each of the Points in the Arc CD, on the Chord CD, and set thereto the Numbers 10, 20, &c. and the Line of Longitude will be divided.

The Reason of this Construction is, that As Radius is to the Sine Complement of any Latitude, So is the Length of a Degree of Longitude under the Equator; which is 60 Miles, to the Length of a Degree of Longitude in that Latitude.

These being all the Lines commonly put upon the Rulers, called Plain-Scales, excepting equal Parts; therefore I shall proceed to shew their manner of using in Trigonometry, and Spherical Geometry.

But, by the way, note, That Plain-Scales are commonly of these two Lengths, *viz.* some one foot long, and others, which are put into Cases of Instruments, but half a foot in Length; and on one Side is a Diagonal Scale: they are generally made of Box, and sometimes of Brass or Ivory.