fer the faid Divifions to the Chord Line AB, and fet thereto the Figures 10, 20, 30, &c. and the Line of Chords AB will be divided, and then may be put upon your Scale.

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 the Numbers 10, 20, &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 Divifion of the Arc BD, draw right Lines cutting BF, which will divide it into a Line of Tangents, fetting 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 fetting thereto the Numbers 10, 20, 30, &c. the Line EG will be divided into a Line of Secants.

To project the Semitangents, draw Lines from the Point C thro' every Degree of the Quadrant AB, and they will divide the Semidiameter AE into a Line of Semitangents: But because the Semitangents on Scales 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 faid continued Portion of AE, and you will have a Line of half Tangents to 160 De grees, or farther if you pleafe,

B

Moreover,

Moreover, to draw the Rhumb Line from every 8th Part of the Quadrant AC, fetting one Foot of your Compaffes in A, defcribe an Arc cuting the Chord AC, which will divide AC into a Line of whole Rhumbs; and after the fame manner may the Subdivifions of half and quarter Rhumbs be made.

Laftly, to project the Line of Longitude, draw. the Line HD equal and parallel to the Radius CE; which divide into 60 equal Parts, every 10 of which number. 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 Compaffes in D, transfer the Distances from D to each of the Points in the Arc CD, on the Chord CD, and fet thereto the Numbers 10, 20, &c. and the Line of Longitude will be divided.

These be all the Lines commonly put on one Side of the Plain-Scale, excepting equal Parts, which want no Defcription: And on the other Side is a decimal or diagonal Scale, on which an Inch, or fome Part thereof, as I half, or 1 fourth, is divided into 100 equal Parts, by means of Di-· agonals, in the following manner; Suppofe AB to be the Length of the Scale,

Fig. 3.

which let be 6 Inches; then having chofen a convenient Breadth, as BC, make the Parallelogram AEBC, and draw Lines from the Divifion of every Inch, parallel to BC; then divide an Inch, fuppofe DB, into 10 equal Parts, as alfo FC, and draw the Diagonals D, 10, 10, 20, 20, 30, &c. Likewife divide BC, and AE, sach into to equal Parts, thro' which draw. 19

Parallels,

Parallels, and fet the Numbers as per Figure, and your Scale will be made. After the fame Manner may one half or one quarter of an Inch be decimally divided.

Now if you have a mind to take any Part of an Inch, fuppofed to be divided into 100, as the 48th Part, look for the Divifion 40 on the Line DB, and guiding your Eye along its Diagonal, mark where the 8th Parallel to BD cuts the faid Diagonal, and the Diftance of that Point of Section from the Line DF, will be the 48th Part of an Inch. Again, to find the 100th Part of an Inch, look where the Parallel 1 cuts the Diagonal D 10, and the Distance from that Point to DF will be the 100th Part of an Inch. For from the Similiarity of Triangles, demonftrated per Prop. 4. lib. 6. Euclid. as DF is to r of DB, or 10 F, fo is ro of DF to ros of ᎠᏴ,

..

Hence it is manifeft, that by means of this Scale, any Line may be drawn whofe Length is expreffible in lefs than four Figures, whether they be whole Numbers, or mixed, that is, decimal Fractions and whole Numbers; as, if the Length of the Line be 365, then you must call Inch 100, and fo 3 Inches will be 300. Then the 65 must be taken from DH to the Interfection of the 60th Diagonal with the 5th Parallel, and the Extent from 3 Inches to that Point of Interfection, will give the Length of the Line expreffible by 365. Likewife the fame Extent may be taken for the Length of a Line expreffible by 365, in fuppofeing Inch to be 10. Alfo it may be expreffed

by 3.65, in taking 1 Inch for 1. And lastly, it may be expreffed by .365, in taking 1 Inch for. And this is the Ufe of the Diagonal Scale.

As for the Ufes of the Chords, Sines, Tangents, &c. the Chords are to lay off the Quan tity of any Angle defired upon a Point given, in a right Line. And contrariwife, to measure the Quantity of an Angle already laid down. The firft is done by always taking the Extent of 60 Degrees of Chords between your Compaffes, and defcribing an Arc about the angular Point; and then laying off the Nuinber of Degrees propofed upon the faid Arc, and drawing a right Line from the angular Point. And the latter, by always making an Arc of 60 Degrees of Chords about the angular Points, and then takeing the Chord of the faid Arc between your Compaffes, and measuring it on the Line of Chords.

Example, To make an Angle of 30 Fig. 4. Degrees on the Point A, take 60 Degrees of Chords in your Compaffes, and fetting one Foot of your Compaffes in the Point A, defcribe the Arc DC; then take 30 Degrees from your Chords, and lay them off from D to C, and draw the Line AC, and the Angle CAB will be 30 Degrees.

Now to measure an Angle, fuppofe CAB, take 60 Degrees of Chords between your Com. paffes, and fetting one Foot in the Point A, defcribe the Arc CD; then take between your Compaffes the Distance from C to D, which meafur'd on the Chords will be found to reach

to

to 30 Degrees, the Quantity of the Angle fought.

The Sines are to orthographically project the Sphere, &c.

The Tangents, half Tangents, and Secants, are used in finding the Centers and Poles of projected Circles, in the Stereographical Projection of the Sphere, &c.

The Rhumbs are to lay down the Angles of a Ship's Way in Navigation.

And the Line of Longitude determines, by Infpection, how many Miles there are in a Degree of Longitude, in each feveral Latitude: As, in the Latitude of no Degrees, that is, under the Equator, 60 Miles make a Degree; in. the Latitude of 40 Degrees, 40 Miles make a Degree; in the Latitude of 60 Degrees, 30 Miles make a Degree; in the Latitude of 80 Degrees, 10 Miles make a Degree.

CHAP.