PRACTICAL GEOMETRY.
DEFINITIONS.[55]
A line is perpendicular to another when it inclines not more on the one side than on the other, the angles on both sides being equal.
Parallel lines are those which have no inclination to each other, being everywhere equi-distant, however far produced, or extended.
An angle is the inclination, or opening of two lines, which meet in a point called the vertex, or angular point: and the two lines are called the legs, or sides of the angle.
The measure of an angle is estimated by the number of degrees contained in the arc between its two legs.
A rectilinear angle has its legs or sides, right, or straight lines.
A curvilinear angle has its legs curves.
A right angle is formed by one line perpendicular to another; the measure of which is an arc of 90°.
An acute angle is less than a right angle, or than 90°.
An obtuse angle is greater than a right angle.
An oblique angle may be either acute, or obtuse.
The circumference, or periphery of a circle is the curved line which bounds it, being everywhere equally distant from the centre. The circumference is supposed to be divided into 360 degrees (marked thus °); each degree into 60 minutes, each minute (′) into 60 seconds (″).
An arc is any part of the circumference of a circle.
A chord, or subtense, is a right line joining the extremities of an arc.
The radius of a circle is a right line drawn from the centre to the circumference.
The diameter of a circle is a right line drawn through the centre, and terminated by the circumference.
A semicircle (180°) is that part of a circle which is contained between the diameter, and half the circumference.
A quadrant is the fourth part of a circle, being contained between two radii, and an arc of 90°.
A segment is that part of a circle which is cut off by a chord.
A sector is that part of a circle contained between two radii, and an arc.
A secant is a line which cuts a circle, lying partly within, and partly without it.
A tangent is a line which touches a circle, or curve, without cutting it.
The point of contact is where a tangent touches an arc.
Triangles are figures having three sides, and three angles.
An equilateral triangle has its three sides equal.
An isosceles triangle has only two equal sides.
A scalene triangle has all its sides unequal.
A rectangular, or right-angled triangle has one of its angles a right one, or 90°; and the square of the side opposite the right angle is equal to the sum of the squares of the sides containing that angle; hence a triangle, having its sides proportional to the numbers 3, 4, 5, will be right-angled.
The hypothenuse is the side opposite the right angle in a rectangular triangle.
An obtuse-angled triangle has one of its angles obtuse.
An acute-angled triangle has all its angles acute.
The three angles of any triangle, taken together, are equal to two right angles, or 180°.
The difference of the squares of two sides of a triangle is equal to the product of their sum and difference.
The sides of a triangle are proportional to the sines of their opposite angles.
Quadrangles, or quadrilaterals, are plane figures bounded by four right lines.
A square is a quadrilateral having all its sides equal, and all its angles right angles. The diagonal of a square is equal to the square root of twice the square of its sides: and the side of the square is equal to the square root of half the square of its diagonal.
The diagonal is a right line drawn across a quadrilateral figure, from one angle to another. The sum of the squares of the two diagonals of every parallelogram is equal to the sum of the squares of the four sides.
A parallelogram is a quadrilateral, whose opposite sides are parallel.
A rectangle is a parallelogram having four right angles.
A rhomboid is an oblique-angled parallelogram.
A rhombus, or lozenge, is a quadrilateral, whose sides are all equal but its angles oblique.
A trapezium is a quadrilateral, which has none of its sides parallel to each other.
A trapezoid is a quadrilateral, which has only two of its sides parallel.
Polygons are plane figures bounded by more than four sides.
A regular polygon has all its sides, and angles equal.
The perimeter of a figure is the sum of all its sides.
To bisect—is to divide into two equal parts.
To trisect—is to divide into three equal parts.
To inscribe—is to draw one figure within another, so that all the angles of the inner figure touch either the angles, sides, or planes of the external figure.
To circumscribe—is to draw a figure round another, so that either the angles, sides, or planes of the circumscribing figure touch all the angles of the figure within it.
LINES, ANGLES, AND FIGURES.
To divide a given right line into two equal parts.
From the extremities of the line as centres, and with any opening in the compasses, greater than half the given line, as a radius, describe arcs intersecting each other above, and below the given line. A line being drawn through these intersections will divide the given line into two equal parts.
An arc of a circle is bisected in the same manner.
To bisect an angle.
From the angular point, measure equal distances on the two lines (forming the angle), and from these points, with the same distance as radius, describe arcs intersecting each other. A line drawn from their intersections to the angular point will bisect the angle.
To erect a perpendicular.
From the point A set off any length 4 times to C; from A as a centre with 3 of those parts describe an arc at B, and from C with 5 of them cut the arc at B. Draw A B, which will be the perpendicular required. Any equimultiples of these numbers, 3, 4, 5, may be used for erecting a perpendicular. Plate 2, [Heights and Distances], and Practical Geometry, [Fig. ½].
To erect a perpendicular.
Set off on each side of the point A, any two equal distances, A D, A E. From D and E as centres, and with any radius greater than half D E, describe two arcs intersecting each other in F. Through A, and F draw the line A F, and it will be the perpendicular required.
[Fig. 1.]—Plate, Practical Geometry.
To let fall a perpendicular.
From D as a centre, and with any radius, describe an arc intersecting the given line. From the points of intersection C, and E, with any radius greater than half, describe two arcs, cutting each other at F. Through D, and F draw a line, and D F will be the perpendicular required. [Fig. 2].
To draw a line parallel to a given line.
From any point D in the given line with the radius D C, describe the arc C E, and from C with the same radius describe the arc D F. Take E C, and set it off from D to F. Through C, and F draw C F for the parallel required. [Fig. 3].
To divide an angle into two equal parts.
From B as a centre with any radius describe an arc A C. From A, and C with any radius describe arcs intersecting each other in D. Then draw B D, and it will bisect the angle. [Fig. 4.]
PRACTICAL GEOMETRY.
Fig. 1-9.
Fig. 10-14.
To divide a right angle into three equal parts.
From B as a centre with any radius describe the arc A C. From A with the radius A B cut the arc A C in D, and with the same radius from C cut it in E. Then through the intersections D, and E draw the lines B D, B E, and they will trisect, or divide the angle into three equal parts. [Fig. 5.]
To find the centre of a circle.
Draw any chord A B, and bisect it by the perpendicular C D. Divide C D into two equal parts, and the point of bisection O will be the centre required. [Fig. 6.]
To describe an equilateral triangle.
From the points A, B, as centres, and with A B as radius, describe arcs intersecting each other in C. Draw C A, C B, and the figure A B C will be the triangle required. [Fig. 7.]
To describe a square.
From the point B, draw B C perpendicular, and equal to A B. On A, and C, with the radius A B, describe arcs cutting each other in D. Draw the lines D A, D C, and the figure A B C D will be the square required. [Fig. 8.]
To inscribe a square in a circle.
Draw the diameters A B, C D perpendicular to each other. Then draw the lines A D, A C, B D, B C; and A B C D will be the square required. [Fig. 9.]
To inscribe an octagon in a circle.
Bisect any two arcs A C, B C of the square A B C D in G, and E. Through the points G, and E, and the centre O draw lines, which produce to F, and H. Join A F, F D, D H, &c. and they will form the octagon required. [Fig. 9.]
On a line to describe all the several polygons, from the hexagon to the dodecagon.
Bisect A B by the perpendicular C D. From A as a centre, and with A B as a radius, describe the arc B E, which divide into six equal parts; and from E as a centre describe the arcs 5 F, 4 G, 3 H, &c. Then from the intersection E as a centre, and with E A as a radius, describe the circle A I D B, which will contain A B six times. From F in like manner as a centre, and with F A as radius, describe the circle A K L B, which will contain A B seven times; and so on for the other polygons. [Fig. 10.]
To inscribe in a circle an equilateral triangle.
From any point D in the circumference as a centre, and with the radius D O of the given circle, describe an arc A O B cutting the circumference in A, and B. Through D, and O draw D C. Then, join A B, A C, B C; and the figure A B C will be the triangle required. [Fig. 11.]
To inscribe a hexagon in a circle.
Bisect the arcs A C, B C in E, and F, and join A D, D B, B F, &c., which will form the hexagon. Or carry the radius six times round the circumference, and the hexagon will be obtained. [Fig. 11.]
To inscribe a dodecagon in a circle.
Bisect the arc A D of the hexagon in G, and A G being carried twelve times round the circumference, will form the dodecagon. [Fig. 11.]
To inscribe a pentagon, hexagon, or decagon, in a circle.
Draw the diameter A B, and make the radius D C perpendicular to A B. Bisect D B in E. From E as a centre, and with E C as radius, describe an arc cutting A D in F. Join C F, which will be the side of the pentagon, C D that of the hexagon, and D F that of the decagon. [Fig. 12.]
To find the angles at the centre, and circumference of a regular polygon.
Divide 360 by the number of the sides of the given polygon, and the quotient will be the angle at the centre; and this angle being subtracted from 180, the difference will be the angle, at the circumference, required.
Table, showing the angles at the centre, and circumference.
| Names. | No. of | Angles | Angles at |
| sides. | at centre. | circumference. | |
| Trigon | 3 | 120° | 60° |
| Tetragon | 4 | 90° | 90° |
| Pentagon | 5 | 72° | 108° |
| Hexagon | 6 | 60° | 120° |
| Heptagon | 7 | 51° 25 5′ 7 | 128° 34 2′ 7 |
| Octagon | 8 | 45° | 135° |
| Nonagon | 9 | 40° | 140° |
| Decagon | 10 | 36° | 144° |
To inscribe any regular polygon in a circle.
From the centre C draw the radii C A, C B, making an angle equal to that at the centre of the proposed polygon, as contained in the preceding table. Then the distance A B will be one side of the polygon, which, being carried round the circumference the proper number of times, will complete the polygon required. [Fig. 13.]
Fig. 15-20.
To circumscribe a circle about a triangle.
Bisect any two of the given sides, A B, B C by the perpendiculars E F, D F. From the intersection F as a centre, and with the distance of any of the angles, as a radius, describe the circle required. [Fig. 14.]
To circumscribe a circle about a square.
Draw the two diagonals A C, B D intersecting each other in O. From O as a centre, and with O A, or O B, as a radius, describe the required circle. [Fig. 15.]
To circumscribe a square about a circle.
Draw the two diameters A B, C D perpendicular to each other, through the points A, C, B, D, draw the tangents E F, E G, G H, F H, and E G H F will be the square required. [Fig. 16.]
To reduce a map, or plan, from one scale to another.
Divide the given figure A C by cross lines, forming as many squares as may be thought necessary. Draw a line E F, on which set off as many parts from the scale M, as A B contains parts of the scale N. Draw E H, and F G perpendicular to E F, and each equal to the proportional parts contained in A D, or B C. Join H G, and divide the figure E G into the same number of squares as the original A C. Describe in every square what is contained in the corresponding square of the given figure; and E F G H will be the reduced plan required. The same operation will serve either to reduce, or enlarge any map, plan, drawing, or painting. [Fig. 17.]