GEOMETRY,

IS the fcience of quantity, extention or magnitude; and may be confidered as the art of reducing unknown to known quantities, and of comparing different lengths, fuperficies, and folids together.

*

GEOMETRICAL PROBLEMS.

To form a right angle or erect a perpendicular line.

Set one foot of the compaffes or dividers in a given point; as, e, form an arch c bd; then with one foot in d, bring the other to b, form an arch a c, and with one foot in c cross the arch at a, and a line from a to e is the perpendicular, which forms the right angle at e.

Or with convenient radius, from any (giv en) point above the base, as a, for centre, form an arch croffing the base, bc c; from centre c, form an arch at e or d, and from centre b, cross the arch at e or d, and draw a line from a, as before for the perpendicular required.

Perpendiculars differently erected.

h

EXAMPLES,

a

Any thing whatever is its own perfect measure, yet its contents can not be comprehended or comparatively valued until it be reduced to some known quantity; as inches, feet, miles, acrs, gallons, pounds, &c. each of which has by use acquired a definite meaning; and though we may have a more accurate conception of the size of a thing by sight than by hearing of its contents in such kuown quantities; yet, its value, weight, &c. cannot be so accurately computed, as by comparison with that whose value, weight, &c. is known.

To defcribe parallel lines. Form an arch of the fame fize at each end of a base, and draw a line from the top of one arch to that of another, for the parallel required.

To defcribe an ellipfe or oval. Form a fmall circle and fet one foot of the compaffes in one fide of the circle, with the other form the oppofit side of the oval; fet one foot in the part nearest to the ends of the oval to form the ends.

Or draw parallel lines across each other and defcribe arches as before from the points of meeting till they interfect.

Or carry a ftring with the ends tied together round two fixed points; as a b.

To defcribe a circle which will pass through any three points, (as a b d) not in a straight line. Set one foot of the dividers in each pein: and form, an arch fo that the arches 4 od will point to the centre of the circle required; as, c. In the fame manner the centre of any given circle may be found.

semi.

To measure or form angles, by the divifions of a circle; as in T. 14. Extend the dividers 60° on any circle, or line of chords, as from 0 20. too; fet one foot in the given 10. point, as b, with the other form an arch gf, and take the distance between the lines on the arch and apply it to the circle or line of chords, which will how the number of degrees the angle contains, as 40°. Subtract two angles of a triangle from 180° and the remainder is the third angle; as 40+95 ·-180-45° e.

Any angle may be formed by taking the given number

h

23497,8,9

Line of equal parts 1,625 in

40

1

of degrees; as 95° i in the dividers and fetting one foot in the arch or sweep of 60 as at g, then the other foot will reach to the place where the line ei fhould be drawn from the given point i to form the angle required, 95°.

Rright lines are measured by the equal divifions of a right line, and circular or curve lines by equal parts of a circle or by (the unequal parts of a right line) a line of chords, as from 0 to 180 in the preceeding femicircle.

TRIGONOMETRY,

By Geometrical Conftruction,

Is the art of protracting and measuring, with fuitable inftruments the fides, angles, &c. of plane triangles.

Three parts of a triangle must be given; viz. the three fides ; two fides and one angle, or one fide and two angles to find the parts required. In a right triangle one angle is always given; viz. 90°

Two fides and an angle between them given, to find the other fide, angles, perpendicular and contents.

RULE. Draw one of the given fides; as a b 6,5, then take the length of the other given fide in the dividers, as b c 3,75, and place one foot in the given point, b, with the other form an arch, cd, lay out the given angle; as b 80o 30', and draw the line to the arch cd from thence to the point a7, is the fide required; the fhorteft diftance from the point c to the bafe is the perpendicular ce, 3,7; measure the angles a cas before directed. The area may be found as under multiplication by multiplying the bafe and half the perpendicular together.

per.

23,7 1,85 x 6,5 12,025 area:

35

6,5

30°30

One fide and two angles given to find the other parts.

C

RULE. Draw the given fide its proper length, as a b7; then lay out the angles from their proper points; as a 30,

*The angle given should be between the two given sides, otherwise there may be a mistake made, as from e to d; (for it is as far from b to one as to the other,) so we should be at a loss to know whether the side a c were 4,2 or 7, except the place of its touching the arch were determined, as at d or c.

b 93° and draw lines till they meet in the other point and form the angle required; the length of thefe lines are the fides required. The angle at b, being larger than a right angle obliges the oppofit fide to be used as a bafe and the fhorteft diftance from 6 to the oppofit fide is the perpendicular 3,7. Or lay out as a many degrees from the given point

30°

816

bas the oppofit angle a ore is lefs than 90° for the perpendicular required.

The three fides given, to find the angles, &c.

a 9,5 to b b 4,6 e c7,2 a

RULE. Take the length of one of the given lines; as a 9,5 to b draw a line of this length; a b, then take the length of another given fide; b 4,6 c, in the dividers, fet one foot in its proper point b with the other form an arch de; take the length of the other line c 7,2 a fet one foot in a with the other cross the arch er curve d e, and draw lines from to a and b. The angles may be measured and the contents found as before.

9,5

EXAMPLES..

What is the contents of a stick of timber 8,5 ft long, whofe ends are equal, and each of the following dimenfions, viz. one fide a 2,5 ft. b 1,8 ft and c 2 ft. ?

here 3 fides are given,

What is the area of a piece of land, two fides being meafured viz. 40 Chains and 54 Ch. the angle between the fides 45 ?

If a fhip leaving a port fail S. E. by E. (i. e. S 5 points or 56°15 eaft) 95 miles; what would be her difference of latitude and departure, (i. e. how far fouth and how far eaft would the be from the port ?)

Here two angles are given, viz. 56° 15′ and 90°; for every meridian or degree of longitude cuts each degree or parallel of latitude at right angles, as at b. 56° 15 90 33° 45′ angle required at c from thence a line to b is

dif.of lat.54

56°15'

S.E.byE.95 dift.failed.

33045

departure from meridian
of port 80 miles
dif. of lat. 54

}

A.

To measure, and delineate the fides, anglès, &c. of any given

plane.

RULE. Take the courses with a compafs fuited to that purpose, and measure the distances on each courfe; then on a fixed point; as, a place the centre of a circle,* femicircle or fomething that fhall reprefent the compafs, (or horizon;) and point off the first courfe by the circle as it was taken with the compass, and measure the distance by a scale of equal parts, place a point at the end as b; lay a fcale of equal parts at this point parallel to the fecond courfe on the circle, measure and mark the fecond distance, from this point elay the fcale parallel to the third courfe, meafure and proseed as before through the whole.

When a circle of brass, paper, &c. is used as above fit should remain fixed till all the courses are taken..