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Sines, Tangents and Secants, on the Plane Scale.

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the Divisions in the Quadrant B L to the Right Line B D ; then
is B D a Line of Chords. - -
4thly. From the Points -10, 20, 30, &c. in the Quadrant B D,
draw Right Lines parallel to C D till they cut the Radius C B ;
then is the Line C B divided into a Line of Sines, which must be
numbered from C towards B.
5thly. If the same Line of Right Sines be numbered from B to-
wards C, it will become a Line of versed. Sines ; which may be con-
tinued to 280°, if the same Divisions be transferred on the other
Side of the Centre C.
6thly. From the Centre C, through the several Divisions in the
Quadrant B D, draw Right Lines till they cut the Tangent B T ;
so will the Line BT become a Line of Tangents.
7thly. Setting one Foot of the Compasses in C, extend the other
to the several Divisions, 10, 20, 30, &c. in the Tangent Line BT,
and transfer these Extents severally into the Right Line C S, then
will the Line C S be a Line of Secants.

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furing of Angles according to the common Divisions of the Mariner's Compass. If the Radius A C be divided into 100 or Ioco, &c. equal Parts, and the Lengths of the several Sines, Tangents, and Secants, corresponding to the several Arches of the Quadrant, be measured thereby, and these Numbers be set down in a Table, cach in its proper Column, you will by this Means have a Triangular Canon of Numbers, by which the several Cases in Trigonometry may he resolved. The Right Lines graduated as above, being placed severally upon a Ruler, form the Instrument called the Plane Scale; by which the Lines and Angles of all Triangles may be measured. All Right Lines (as the Sides of Plane Triangles, &c. when they are considered simply as such, without having any Relation to a Circle) are measured by Scales of cqual Parts; one of which is subdivided equally into Io, and this serves as a common Division to all the rest. In most Scales an Inch is taken for a common Measure, to determine their Largeness and Number of Parts: what an Inch is divided into, is generally set at the End of the Scale, as in the Scales A, B, and C3 the Numbers 10, 20, 30, shew that so many Parts of the Scales A, B, C, are contained in an Inch. B any Scale of equal Parts divided as above, any Number less than ico may be readily taken ; but if the Number should consist of Three Places of Figures, the Value of the Third Figure can only be guessed at: wherefore, in these Cases, it is better to use such a Scale as D, called a Diagonal Scale, by which any Number of Three Figures may be exactly found.

Having prepared a Ruler of convenient Breadth for your Scale, (which may be an Inch more or less). First, near the Edges thereof, draw Two Right Lines of, cg, parallel to each other ; then divide one of these Lines as af, into equal Parts, according to the Largeness you intend your Scale; and through each of these Divisions draw perpendicular Right Lines as far as the Line cg ; next divide the Breadth into 10 equal Parts; and through each of these Divisions

draw Right Lines parallel to the former a f and c g; again, divide

the Length, a, b, c, d, each into 10 equal Parts; and from the Point d to the first Division in the Line A B, draw a Right Line; then, parallel to that Line, draw Right Lincs through all the other Divisions, and the Scale is done.

Besides the Lines already mentioned, there is another on the Plane Scale marked ML, which is joined to a Line of Chords; and shews how many Miles Easting or {. make a Degree of Longitude in every Latitude ; these several Lines are generally put on one Side of a Ruler 2 Feet long ; and on the other Side are laid down a Scale of the Logarithms of the Sines, Tangents, and Numbers, which is commonly called Gunter's Scale; and as it is of general Use, it requires a particular Description.

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J lowing. 1st. Sine Rhumbs (marked S R) is a Line which contains the Logarithm of the Sine of every Degree, Point, and Quarter Point of the Mariner's Compass, figured from the Left Hand towards the Right, with 1, 2, 3, 4, 5, 6, 7, to 8, where is a Brass Pin, and where it can, it is divided into Halves and Quarters. 2d. Tangent Rhumbs (marked TR) also corresponds to the Logarithm of the Tangent of every Degree of the said Compass, and is figured 1, 2, 3, 4, at the Centre, where is a Pin, and from thence towards the Left Hand with 5, 6, 7, it is also divided, where it can, into Halves and Quarters. 3d. The Line of Numbers (marked Num.) contains the Logarithm of the Numbers, and is figured thus; near the Left Hand End it begins at 1, and towards the Right Hand is 2, 3, 4, 5, 6, 7, 8, 9 ; then I is the Middle, at which is a Brass Centre Pin, going still on 2, 3, 4, 5, 6, 7, 8, 9, and 10 at the End, where is another Centre Pin: (As this Line is generally used, it requires a larger Description) The first I may be counted for 1, or 10, or Ioo, or Iooo, and then the next 2 is accordingly 2, or 20, or 200, or 2000, &c. Again, , the first 1 may be reckoned for 1 Tenth, or 1 Hundreth, or I Thousandth Part, &c. then the next is 2 Tenth, or 2 Hundredth, or 2 Thousandth Parts, &c. so that if the first 1 be esteemed I, the Middle I is then IQ, and 2 to its Right is 20, 3 is 30, 4 is 40, and 10 at the End is 100 ; again, if the First 1 is 10, the next 2 is 20, 3. is 30, and so on, making the Middle I now Ioo, the next 2 is 200, 3 is 300, 4 is 400, and 10 at the End is now Iooo. In like Manner if the First I be esteemed 1 Tenth Part, the next is 2 Tenth Parts, and the Middle 1 is 1, and the next 2 is 2, and 10 at the End is now Io. Again, if the First I be counted 1 Hundredth Part, the next is 2 Hundredth Parts, the Middle 1 is now 10 Hundredth Parts or 1 Tenth Part, and the next 2 is 2 Tenth Parts, and 10 at the End is now but one whole Number or Integer. As the Figures are increased or diminished in their Value, so in like Manner must all the intermediate Strokes or Subdivisions be increased or decreased ; that is, if the First 1 (at the Left Hand) be counted 1, then 2 (on the Right Hand of it) is 2 ; and each Subdivision between them now is 1 Tenth Part, and so all the Way to the Middle J, which now is 10, the next 2 is 20 ; now the t longer

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