along the bottom; then, for the 3, flide up both points of the compaffes by a parallel motion, till they fall upon the third longitudinal line; and in that pofition extend the fecond point of the compaffes to the fourth diagonal line, and you have the extent of three figures as required.

Or if you have any line to measure the length of. -Take it between the compaffes, and applying it to the fcale, fuppofe it fall between 3 and 4 of the large divifions; or, more nearly, that it is 3 of the large divifions, or 3 hundreds, and between 5 and 6 of the fecond divifions, or 5 tens or 50, and a little more. Slide up the points of the compaffes by a parallel motion, keeping one foot always on the vertical divifion of 3 hundred, till the other point fall exactly on one of the diagonal lines, which fuppofe to be 8, which is 8 units. Which fhews that the length of the line, proposed to be measured, is 358.

The above are three other forms of scales, the first of which is a decimal scale, for taking off common numbers confifting of two figures. The other two are duodecimal fcales, and ferve for feet and inches, &c.

These, and other fcales, engraven on ivory, are fitteft for practical ufe. And the most convenient

D 2

form

form of a plane fcale of equal divifions, is on the very edge of the ivory, made thin at the edge, for laying along any line, and then marking on the paper oppofite any divifion required: which is better than taking lengths off a fcale with compaffes.

2. The chord AC is a mean proportional between AD and the diameter AB. And the chord BC a mean proportional between DB and AB.

That is, AD AC :: AC: AB,

and BD BC BC: AB.

3. The angle ACB, in a femicircle, is always a right angle.

4. The fquare of the hypotenufe of a right-angled triangle, is equal to the fquares of both the fides. That is, ACAD2 + DC2,

5. Triangles that have all the three angles of the one, refpectively equal to all the three of the other, are called equiangular triangles, or fimilar triangles.

6. In fimilar triangles, the like fides, or fides oppofite the equal angles, are proportional.

7. The areas, or fpaces, of fimilar triangles, are to each other, as the fquares of their like fides.

SECT.

SECT. II.

PLANE

TRIGONOMETRY.

LANE Trigonometry teaches the relations and calculations of the fides and angles of plane

PLA

triangles.

The angles of triangles are measured by the number of degrees contained in the arc cut off by the legs of the angle, and whofe center is the angular point. A right angle is therefore an angle of 90 degrees; and the fum of the three angles of every triangle, or two right angles, is equal to 180°. Wherefore, in a right-angled triangle, the one acute angle being fubtracted from 90°, the remainder will be the other; and the fum of any two angles of a triangle being taken from 180°, will leave the third angle.

Degrees are marked at the top of the figure with a finall, minutes with ', feconds with ", and so on. Thus 57° 30′ 12′′, that is, 57 degrees 30 minutes and 12 feconds.

In a right-angled triangle, the fide oppofite the right angle, is called the hypotenufe; and the other, the legs, or fides, or fometimes the base and perpendicular.

The complement of an arc is what it wants of a quadrant. So BC 40° is the complement of

AB = 50°.

The supplement of an arc is what it wants of a femicircle. So BCD 130° is the complement of AB 50°.

The fine of an arc is the line drawn from one end of the arc perpendicularly upon the diameter drawn through the other end of the arc. SO BE is the fine of AB or of BCD.

The verfed fine of an arc is the part of the diameter between the fine and the beginning of the arc. So AE is the verfed fine of AB, and DE the verfed fine of BCD.

The tangent of an arc is the line drawn perpendicularly from one end of the diameter paffing through one end of the arc, and terminated by the line drawn from the center through the other end of the arc. So AG or DK is the tangent of AB, or of

BCD.

The fecant of an arc is the line drawn from the center through the end of the arc, and terminated by the tangent. So FG or FK is the fecant of AB, or of

BCD.

The cofine, cotangent, or cofecant of an arc, is the fine, tangent, or fecant of the complement of that arc. So BH is the cofine, ci the cotangent, and FI the cofecant of AB.

From the definitions it is evident that the fine, tangent, and fecant, are common to two arcs, which are the fupplements of each other: fo the fine, tangent, or fecant of 50°, is also the fine, tangent, or fecant of 130°.

The fine, tangent, or fecant of an angle, is the fine, tangent, or fecant of the arc, or the degrees, by which the angle is measured.

The fine, tangent, and fecant of every degree and minute in a quadrant, are calculated to the radius 1, and ranged in tables for ufe. But because trigonometrical operations with thefe natural fines, tangents, and fecants, require tedious multiplications and divifions, the logarithms of them are taken, and ranged in tables alfo; and the logarithmic fines, tangents, and fecants are commonly ufed, as they require only additions and fubtractions, inftead of the multiplications and divifions.

The method of conftructing the fcales of chords, fines, tangents, and fecants, ufually placed on inftruments, is exhibited in the following figure.

There are ufually

three methods of refolving triangles, or the cafes of trigonometry ; namely, Geometrical conftruction, Arithmetical computation, and Inftrumental operation.

In the firft method, let the triangle be conftructed, by making the parts of the given magnitudes, namely the fides from a fcale of equal parts, and the angles from a fcale of chords, or other inftrument. Then meafure the required parts by the fame fcale.

In the fecond method,

having stated the terms

of the proportion ac

cording to the rule, refolve it like all other propor