PROBLEM XVIII.

To divide a Circle into any Number of equal even Parts, as 4, 16, 32.

First draw the diameter through the centre, which will divide it into two equal parts; bifect the diameter with another right line perpendicular thereto, and the circle will be divided into four equal parts or quadrants; bisect each of these quadrants again by right lines drawn through the centre, and it will be divided into eight equal parts, and fo may you continue on your bifections any number of times, that is 4, 8, 16, 32, &c. doubling the number of even parts.

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This Problem is useful in constructing the Mariner's Compass.

I. A chord or fubtense of an arch, is a right line that divides the circle into two unequal parts, and is

a chord to them both, as FH, TI.

II. A right fine of an arch is a line drawn from the end or termination of an arch, perpendicular to the radius, or is half the chord of twice the arch, fo that TV is the fine of the arch TG, and of the arch TF, the fum of which arches together make 180°, or a femi-circle.

III. The verfed fine of an arch is part of the diameter intercepted between the right fine and the arch, as VG.

F

H

S

IV. The tangent of an arch is a line drawn perpendicular to the end of the radius, or diameter, juft touching the arch, as DG.

V. The fecant of an arch is a right line drawn from the centre through the circumference, meeting the end of the tangent line to the fame arch, as OD is the fecant of the arch TG, to which DG is tangent; alfo OR is the fecant of the arch CT, to which CR is a tangent.

NOTE. Sines, Tangents, Secants, are faid to be the measure of fo many degrees as the arch contains parts of 360, fo that radius being the fine of a quadrant, or a fourth part of the circumference, contains 90 degrees; thus the radius is always equal to the fine of 90°, as is alfo the tangent of 45°, and the chord of 60°.

B 2

PROJECTION.

PROJECTION

OF THE LINES OF

SINES, TANGENTS, AND SECANTS,

ON THE PLANE SCALE.

1ft. We

7ITH the radius you intend for your fcale, defcribe a femi-circle ADBC, and upon the centre C raise the perpendicular CD, (which will divide the femi-circle into two quadrants, AD, BD), continue CD directly to S, and upon B raise the perpendicular BT, then draw the right lines BD and AD.

2dly. Divide the quadrant BD into 9 equal parts, then will each of these be 10 degrees. Again, you may fubdivide each of these parts into fingle degrees; and these again, if your radius admits it, into minutes, or fome aliquot parts of a degree greater than minutes.

3dly. Set one foot of the compaffes in B, and transfer each of the divifions in the quadrant BD to the right line BD, then is BD a line of chords.

4thly. From the points 10, 20, 30, &c. in the quadrant BD, draw right lines parallel to CD, till they cut the radius CB, then is the line CB divided into a line of fines, which must be numbered from C towards B.

5thly. If the fame line of right fines be numbered from B towards C, it will become a line of verfed fines, which may be continued to 180°, if the fame divifions be transferred on the fame line on the other fide of the centre C.

6thly. From the centre C, through the several divifions in the quadrant BD, draw right lines till they cut the tangent BT, fo will the line BT become a line of tangents.

7thly. Setting one foot of the compaffes in C, extend the other to the feveral divifions 10, 20, 30, &c. on the tangent line BT, and transfer these extents feverally into the right line CS, then will the line CS be a line of fecants.

8thly. Right lines drawn from A to the feveral divifions, 10, 20, 30, &c. in the quadrant BD, will divide the radius CD into a line of femi-tangents.

9thly. Divide the quadrant AD into eight equal parts, and from A transfer thefe divifions feverally into the line AD, then is AD a line of rhumbs, each divifion answering to 11° 15' upon the line of chords.

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The use of this line is for protracting and measuring of angles, according to the common divifion of the Mariner's Compafs. If the radius AC be divided into 100, or 1000, &c. equal parts, and the lengths of the several fines, tangents, and fecants, correfponding to the several arches of the quadrant be meafured thereby, and these numbers be fet down in a table, each in its proper column, you will, by these means, have a triangular canon of numbers, by which the feveral cafes in Trigonometry may be folved, the right lines, graduated as above, being placed feverally upon a ruler, form the inftrument called the Prane Scale; by which the lines and angles of all triangles may be measured. All right lines, as the fides of plane triangles, &c. when they are confidered fimply as fuch, without having any relation to a circle, are measured by scales of equal parts, one of which is fubdivided equally into 10, and this ferves as a common divifion to all the reft. In moft fcales an inch is taken for a common measure to determine their largeness and number of parts; what an inch is divided into is generally fset at the end of the fcale, as in the fcales A, B, and C; the numbers 10, 20, 30,45, fhew that fo many parts of the fcales A, B, C, are contained in an inch. By any scale of equal parts, divided as above, any number less than 100 may be readily taken; but, if the number should confift of three places of figures, the value of the third figure can only be gueffed at; wherefore, in thefe fcales, 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), firft, near the edges thereof, draw two right lines, af, eg, parallel to each other; then divide one of these lines, as af, into equal parts, according to the largeness you intend your fcale; and through each of thefe divifions draw perpendicular right lines as far as the line c g; next divide the breadth into 10 equal parts, and through each of thefe divifions draw right lines parallel to the former af and cg; again divide the length a, b, c, d, each into 10 equal parts, and from the point to the first divifion in the line d q, draw a right line; then parallel to that line, draw right lines through all the other divifions, and the fcale is done.

Befides the lines already mentioned, there is another on the plane scale, marked ML, which is joined to a line of chords; and fhews how many miles, eafting or wefting, make a degree of longitude in every latitude; these several lines are generally put on one fide of a ruler, two feet long; and on the other fide are laid down a scale of the logarithms of the fines, 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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