GEOMETRICAL DEFINITIONS.

EOMETRY is the Science which treats of the Defcription, Properties, and Relations of Magnitudes in general; of which there are three Kinds or Species, viz. a Line, which has only Length without either Breadth or Thicknefs; a Superficies, comprehended by Length and Breadth; and a Solid, which has Length, Breadth, and Thicknefs.

I.

A point confidered mathematically, is incapable of being divided, and therefore hath no parts, or it is the finalleft part of space that can be affigned, and may be conceived A. fo infinitely fmall, as to be void of length, breadth, or thickness, being always denoted by a dot, as at A.

II.

A right line is the nearest distance between two points, which limits its length, without any fuppofed breadth, or thicknefs, as AB; it A: may be supposed to be the flowing of a point.

III.

B

A plane fuperficies is that which lies evenly between its extreme points, refembling a fmooth table, or polifhed glafs; bounded by lines having length and breadth: but is conceived to have no depth or thickness, and may be conceived to be generated by the flowing of a right line.

IV.

Parallel lines are fuch as are equally distant in all their parts, which extended infinitely on the fame plane would never meet, as the lines АВ, БС.

V.

A plane angle is the inclination or meeting of two right lines in one point; the point where they meet is called the angular point, and the lines AB and AC are called fides or legs; it is A generally expreffed by three letters, the middle one always denotes the angular point, as A, and the other two the legs or fides that include it, as AB or AC.

A

B

VI. A circle

VI.

A circle is a plane figure, bounded by an uniform curve line; it is ordinarily defcribed by a right line, taken with a pair of compaffes; one point thereof being fixed, whilft the other is turned round to the place where the motion first began; the fixed point is called the centre, and the line defcribed by the other point is called the circumference.

An arch of a circle is any part or portion of the circumference, as DFE.

IX.

A chord of a circle is the fubtence of an arch, or it is a right line joining the ends of an arch; it divides the circle into two unequal parts, called fegments, and is a chord to them both, as DE is the chord of the arches DFE and DGE.

X.

A femicircle, or half a circle, is a figure contained under the diameter, as AGB or AFB.

XI.

A quadrant is half a femicircle, or one fourth part of the whole circle; as the figure CAG.

NOTE. All circles, whether great or fmall, are actually, or fuppofed to have, their circumference divided into 360 equal parts, called degrees, and each degree into 60 equal parts, called minutes, and each minute into 60 equal parts, called feconds, and fo on into thirds, fourths, &c.

All angles are measured by an arch of a circle, defcribed round their angular points, with the chord of 60 degrees, taken from the line of chords on the plane fcale, and are estimated greater or less according to the number of degrees contained betwixt their legs; and though legs be ma 'e longer or fhorter, ftill the angle between them continues the fame.

XII.

A right line is faid to be PERPENDICULAR to another line, when it falls upon it fo as to make the angles on each fide of it equal, fuch as the figure ABCD, where the angle ACD is equal to the angle ACB, each a quadrant, or right angle, containing 90 degrees.

XIII.

D

An ACUTE ANGLE is lefs than a right angle, and is that which contains less than 90 degrees, as ABC.

The feweft number of right lines that can include a space are three, which form a figure called a triangle, or three-cornered figure, and confifts of fix parts, viz. three fides and three angles; it is diftinguished into three forts, viz. a right-angled triangle, an obtufe-angled triangle, and an acute-angled triangle.

XV.

A RIGHT-ANGLED TRIANGLE has one of its angles right, or containing 90 degrees; the fide oppofite the right angle is called the hypothenufe, and the other two fides are called legs; that which ftands upright is called the perpendicular, and the other the base: thus BC is the hypothenuke, AC the perpendicular, and AB the bafe; the angles oppofite the two legs are both acute.

XVI.

An ACUTE-ANGLED TRIANGLE has all its angles acute, or none of them equal to 90 degrees, as DEG.

B

E

XVII.

An OBTUSE-ANGLED TRIANGLE has one of its angles obtufe, or greater than go degrees, as RAF, the other two angles are acute, or lefs than 90 degrees, as in the triangle RAF.

A

NOTE. All triangles that are not right-angled, whether they are acute or obtufe, are in general terms called oblique-angled triangles,, without any other diftinction. The fum of the two acute angles of a right-angled triangle make 90°, the fum of all the angles of any triangle 180°. If from 180 you take the fum of the other two angles, the remaining angle will be found; but in a right-angled triangle, if from 90 you fubtract the one angle, the other angle will remain.

MARKS OR CHARACTERS.

Signifies more, or the Sign of Addition; it fhews that whatever numbers or quantity follow this Sign must be added to those that go before it, thus 9+8, that is 9 added to 8. Or, A+B implies that the quantities reprefented by A and B are added. Signifies lefs, and is ufed as the Sign of Subtraction; it denotes that the number following it must be subtracted from those going before it, as 7-5, or 5 fubtracted from 7.

72

12

× The Sign of Multiplication, and thews that the numbers placed before and after are to be multiplied, thus 7×9, that is 7 multiplied by 9, which makes 63, and 7x8x2 which makes 112. This mark ftands for Divifion, and fignifies that the number that stands before it is to be divided by the number following it, as 72-12 fhews that 72 is to be divided by 12. Or thus,, The Sign of Fquality: it fhews that the numbers or quantities placed before it are equal to thofe following it. thus, 8x12=96. Or 8 multiplied by 12 is equal to 96, and 7 +2 × 436. :::: Proportion, and is read thus, 7:14:: 10: 20, that is, as 7 is to 14, fo is 10 to 20. Or, A: B:: C: D, that is, as A is to B, fo is C to D.

• Signifies Degrees, thus 45° fhew the number 45 degrees. 1 Signifies Minutes, thus 24' or minutes.

"Signifies Seconds, thus 44′′ or 44 feconds.

S Stands for Sine.

Sec.for Secant.

Tan.

-Tangent.

Each of thefe laft with Co. before them, fignifies the comple. ment, as Co-fine, Co-tangent, Co-fecant.

Signifies Angle.

2d Angled, with an s at top Angles s

A Signifies Triangle, or As.

Z Is frequently put to fignify the fum of any two lines or numbers.

Y Signifies the difference.

GEOMETRICAL PROBLEMS,

USEFUL IN NAVIGATION.

A PROBLEM is a practical PROPOSITION, in which Something is propofed to be done or effected.

PROBLEM I.

To draw a Right Line parallel to a given Right Line, to any given
Distance, as at the Point D.

7ITH a pair of compaffes take the
nearest distance between the point

WIT

C

D and the given right line AB, with that diftance fet one foot of the compaffes any. A where on the line AB, as at A, and draw

B

the arch C, from the point D draw a line fo as juft to touch the arch C, and it is done; for the line CD will be parallel to the line AB, and at the diftance of the point given D, as was required.

PROBLEM II.

To bifect or divide a given Line into two equal Parts. With any distance in your compaffes greater than half the line AB, with one foot in B, defcribe an arch with the fame distance, and one foot in A, defcribe an arch that will cut the former A arch in C and D; through C and D draw a line, and that will cut AB in E; and the line AB will be divided at the point E into two equal parts.

PROBLEM III.

-B

Torect a Perpendicular on the End of a given Right Line, as DB. With any distance in your compaffes, as

from B to C, with one foot in C, defcribe the circle BDA, fo that it may juft touch the end of the given line at B; from whence the circle cuts the line as at D, draw a line through the points D and C, to cut the circle in A; from A draw the line AB, which will be the perpendicular required.

Or thus,

C

B