Chapter II.
Introductory exercises in drawing with compasses and a ruler.
The first and second plate refer to these exercises. The points, lines, and surfaces, are designated by letters.
I. To draw a right or straight line.
Have a very sharp pointed black lead pencil, and keep uniformly close to the ruler in drawing the line, taking care to have it very fine and al¬most imperceptible, for a line is that which has length without breadth. x
II. To take the length of a line with the compasses.
The compasses must be opened so wide that the ends of the two legs may exactly touch the two extremities, or the beginning and terminating points of a line. The place where a line commences or ends, or where two lines pas* through, or intersect each other, is a point.
III. To take the length of a long line in order to make another exactly similar, as for instance of a line longer than a common size sheet of paper.
Divide the given line, by dots, into several parts, at pleasure; then set off each part succes¬sively with the compasses.
IV. To marie the exact division of a given line on another line.
Whenever the given line is not very long
e i f as for instance, the line o 7-g, [?] this is most accurately done by measuring not the intermediate successive lengths, but .by taking the spaces, be, bd, bfor gf, gd, gc, beginning al ¬ways with b or g.
But when the line is of considerable length, a common ruler or a straight slip of paper may be \ held against it, and the divisions may be marked on the same.
Proposition III. may also be done in a similar manner.
V. To give the distance of a point from an opposite > line, as, for, instance, the distance of the point b • from the line cd. (See plate I. fig. 1.)
One end of a pair of compasses, that have been previously opened nearly as wide as is the dis¬tance of the point b from the line c d, is placed on this point b9 and the other end is made to describe ah arc which touches the line, but does not intersect it. In this case the distance is given by the width between the two ends of the compasses.
9
VI. ' To divide a given line into two equal parti with the compasses, as, for instance, the line f g, or I m (fig. 2, or 3, plate I.)
Set one point of the compasses in k as near the middle of the line ad possible, then open the compasses to g, and keeping one point fixed in k, turn the other towards ƒ. Then if there be any part of the line left, as fh (fig. 2), one point of the eompasses is kept steady on k, and the other is carried by guess to the middle of fh. Should there now be too much, as the dis¬tance I pi (fig. 3), one point of the compasses id kept steady on n, and the other is carried to about the middle of I p nearer the other point of the compasses.
If on examination* the line is not yet divided intatwo equal parts, the same process must be repeated until the two parts are of equal length.
Or it may be done thus:
10
To divide the line fg into two equal parts.
h From the points/andg, as centres, with any opening of the compasses greater than half fg, describe arcs cutting each other in a and b.
2. Braw the line ob, and the point e, where it cuts fg9 will be the middle of the fine required*
VII. To divide a line into more than two equal parts, as, for instance, the line q r (fig. 4, pi. I.) or the line v w (fig. 5, pi. I.) into three.
Take, by guess, as near a third part of the line as possible, then after having set off three equal parts on the liner;; should there be a remainder, as q s -(Fig. 4), one point of the compasses must be kept steady on t, and one-third of q s must be taken by the other. But if, by turning over the compasses three times, the point should fall beyond the line, as » z (fig. 5), ofte point of the compasses must be kept steady on x, and the other must be brought one-third of z v, nearer to e.
The same is to be observed when the line is to be divided into 4, 5, or more equal parts, with this difference, that whatever remains may always be subdivided; as, for instance, when the ques- ' tion is of six parts, into six. ~ And whenever such parts are many, they may first be divided into a smaller number of parts, and then each part agt^n separately; as, for instance, when the question is of six parts, twice 3 or three times 2 is equal to six, therefore the given line may he first divided into two parts, and every part again into three; or first into three parts, and every part again into two. ' ,
Or may be done thus:
To divide & given line as the following 4 A Into any proposed number of equal parts, as, for instance, the line 4 S into ftve.
describe the arc Bm, also with the same opening of the com¬passes, and one point in B, describe the arc An; or with a parallel ruler, draw Be parallel to A c. •
* 2. From one end A, of the line, draw Ac,. to any point, as c, in
the arc Bm, then take the arc Be in the compasses, and s£f^>ff an . equal arc on An, i. e,.Ae, then draw the line Be,
3. In each of the lines Ac, Be, commencing at A and B • set off as many equal parts of any convenient length, as A B is to be divided into.
ƒ 4. Join the points A, 5; 1,4; 2, 3; 3, 2; 4, 1; 5, B; and AB will then be divided into five equal parts as was required.
VIII. To place the length of a given line exactly in the middle of another line, as, for instance, the line b c (fig. 6, pi. I.) on the line d f
Setoff the shorter line and the longer one from d to g, and divide the remainder gf into two equal parts. Each of these parts is the distance at which the extremities of the shorter line are to be from the extremities of the longer, if the shorter line is to be precisely on the middle of the long one. When therefore the distance is taken from th^beginning, as d a, and terminating point, as ƒ e, towards the centre of the longer line, the . middle space between (viz. a c) is the length of the shorter line.
.Supposing the line to be placed in the middle be the width of a door, and the longer one be the length of a small building, the door by this pro¬* cess would be placed exactly in the middle of the
building.
IX. To set off a line more than Once upon another longer one, so that the intervals be equal; as, fir instance, the tine hk (fig. 7> pl.I.) three times on the line I m. %
The length of the shorter lin$ must be set off upon the longer, either from the beginning or
14
terminating point, as often as required; here, of course, three times. The remainder nm must then be divided, into as many equal parts as the number of times the line is to be set off upon the other, less one; here therefore into two parts. After which, first take the shorter line and then
, one of the two equal parts, as intermediate spa&s, beginning precisely at one of the extremities of the longer line.
And the same manner of proceeding takes place when the line is to be set off four, five, six, &c. times. ,
X. To set off a line several times upon a longer one and so that there be not only equal intervals ' but that the line may be at an equal distance from both extremities of the greater line; as, for in¬stance, the linep q (fig. & pi. 1.) three times on
the line r $.
The shorter line p q, as in the preceding case, is to be placed on the longer one r s three times
from r to <: the remainder t s is then divided into as many equal parts as the shorter line is to be placed on the longer, and one more: consequently here into four parts. Each of these four parts is the required distance, and at the same time the intermediate space.
It.is in this way that the proper place is assign¬ed to doors or windows in a building.
XI. To make two lines meet exactly like too others, as for instances, omitting the dotted art (fig. 9. pi. 1.) the lines » c and g b, like tke liries •ox endow.
Draw the line a c, if it be not already 'drawn, and with the same opening of the compasses des¬cribe from the points where they are to meet*, as here from o and s, two small arcs; set off the length of the arc between the aforesaid lines, viz. x w, fromc to b, and draw a line from » to b, and it wiU meet the line a c, as required.
Such a meeting of two lines in one point is called an angle. Whenever a line meets an¬other without having any inclination, it is a right angle, and the line itself in this case is said to be standing right angularly, or perpendicularly, or straight upon the other. ,
When the lines are more inclined to one an¬other than the quantity of a right angle, it is called an acute angle; but when they are less inclined to one another than a right angle, it is an obtuse angle. . Whenever a line is not per¬pendicular upon another, it is inclined.
To describe an arc from a given point, place one end of the compasses on that point, and de¬scribe the arc with the other end.
XII. To draw a perpendicular line upon another, as,t for instance, the line gf upon the line d/ , (fig. 10. pi. 1). /
This is best done with a triangular ruler. Put the comiQon ruler against the line on which. the
other line is to be placed, as here the line df- and bring upon this ruler one of the short sides of the triangular ruler, then holding both rulers firm one against the other, draw the required line along the other side of the triangular ruler.
And. to try this ruler, draw in the aforesaid manner a line at right angles to another, and holding the common ruler firmly, turn the trian-gular ruler upon the right angular line just drawn, and move it on the common ruler exactly against this line. If the edge of the ruler and the line meet again: accurately, the triangular ruler is correct.
XIII. To draw from a given point a line at right angles to another line, without the triangular ruler, as, for instance, from the point h (fig. 11. pi. 1), a line perpendicular to k I.
Describe from the points of intersection k and
/, and draw two other arcs intersecting each
18 .
other at m. Then placing the ruler against the points h and m, draw a line from the point h to the line k I, and it will be at right angles to k I.
When small arcs like those (fig. 11) are described, both above and under a line from the two extreme points or of the same; and when . the said line is intersected by placing the ruler on the two points where the arcs meet, that line will be divided exactly into two equal parts..
XIV. To draw, but without the triangularruler, another line at right angles, on a given point of a line, as, for instance, the line rn on the point n of the line pq (fig. 12. pi. 1).
Place one end of the compasses on the given point n, and describe with the other small arcs intersecting the line, as here at p and q. De¬scribe again small arcs from these points p and q, intersecting each other above the line, as here
19
at r, and then draw the required line by holding the common ruler against the points r and n.
XV. To draw, without the triangular ruler, an¬other line at right angles at the end of a given line, as, for instance, the line h upon the line f (fig. 20, plate 2).
The point ƒ may be taken at pleasure; but de-scribe, with the same opening of the compasses, from this point ƒ and the terminating point of the line, two arcs intersecting each other as here at g. Afterwards draw a line through the points ƒ and g, then make g h equal to gf, and draw a line from the point A- to the aforesaid terminating point.
XVI. To divide an angle into several equal parts, as, for instance, the angle t s v (fig. 13. plate 1.) into four equal parts.
Describe with the compasses, from the angular
point s9 an arc meeting both the lines that in¬close the angle or its sides, as here at t9 and r. Afterwards divide the arc between the two lines of that angle into as many equal parts as the an¬gle is to be divided into, in this case of course into four; and then draw lines through these points of subdivision to the angular point 5.
The larger or smaller the arc tv is, in case of similar distances, as s v, which however may be more or less considerable, the larger or smaller is the angle itself. The length of the lines has no effect on the magnitude of the angle, which varies only according to the inclination of the lines towards each other.
XVII. To draw a line to be every where equidis- tantfrom another, or parallel to another line; as, for instance, the line w x (fig. 14, pi. 2,) pa¬rallel to the line zb.' '
From any two points, at pleasure, as ac, in the line z'b9 with an opening of the compasses equal
to the required distance, describe the arcs n m, and placing the ruler upon the upper extremities of these arcs, draw a line (without cutting them) to touch the two arcs.
Equidistant or parallel lines may be seen, for instance, in the side walls of houses, in doors, windows, and other objects:
XVIIL Another way of drawing one line parallel to another; as, for instance, the line c d to the
. Knefg (fig. 15, plate 2).
Draw on the given line, ƒ g, if possible at a good distance, two lines at right angles and of the same length, and through the extremities of those per-pendiculars draw another straight line. This will be parallel to the given line.
When several lines are required to stand per-pendicular upon another, and consequently pa¬rallel to one another, the shortest way is to put the triangular ruler against the common ruler, and to keep it firm against it. ’
XIX. To draw through a given point opposite to a given line another line parallel to this line, as, for instance, the line h k parallel to the line lm through the point h (fig. 16. pi. 2).
Seek, by means of prop. V., the distance of the point h from the line lm; place one end of the compasses on this line I m very far from h, and describe a small arc opposite to this point. Afterwards put the common ruler against this arc and the aforesaid point h, and draw the in¬tended line.
XX. To take the distance of two parallel lines.
Proceed, either according to prop. V., selecting at pleasure a point on one of the two lines, and describe from this point an arc which closely touches the other line. The opening of . the com¬passes is, in that case, the required distance.
Or, put the ruler against one of the two lines, and draw (which is most readily done with the
triangular ruler) a. line at right angles upon it, so as to intersect the other of the two given lines. Hie length of this right angular line drawn be¬tween the two parallel lines is, in this case, the required distance.
This process determines the length or breadth of objects inclosed within parallel lines, as, for instance, of doors, windows, and such like. With regard to diminutive dbjeets, only the aforesaid ' arc'or the right angular line is to be attended to.
XXI. To make three given line» meet at their ex-tremities, or to form a figure with them ; as, for instance, the lines n,p, q (fig. 17, plate 2).
' Draw r s, a line as long as one of the three given lines, for ex. n: then from both extremities • of the line rs describe, with the separate lengths of the other two, arcs intersecting each other, as at m; and draw two lines from the extremities r and s, to the point of intersection. The figure
may also be begun with either of the other given lines; instead of the line n.
A figure is a space; completely bounded by lines; an angle therefore, as for instance that at fig. 11, is not a figure. But any figure inclosed within three lines is called a triangle.
XXII. To describe an equilateral Triangle, viz. one whose sides are all equal, as, for instancef that (fig. 18., plate 2.) with the given line tv.
Draw the line wx9 as long as t and with an opening of the compasses equal to tv QT W X9 draw small arcs intersecting each other in m, (see plate 2. fig. 18). and from the point m where they inter¬sect each other, draw lines to the points w and x.
XXIII. To describe an Isosceles Triangle, viz. one of which two sides only are equal; as,for instance,' the Triangle (fig. 19, plate 29).with the given lines p and q* .
Draw the line z equal to the line y; with the
length of the line p describe from both extremi¬ties of the line z two $mall arcs intersecting each other in m, and draw lines from their common intersection to the extremities of the line z.
XXIV. To describe a Scalene Triangle, viz. one of which all the three sides are unequal; as, for in¬
' stance, the Triangle (fig. 17, plate 2,) with the
given lines, n p q<
The process is the same as in prop. XXI.
XXV. To describe a Right-angled Triangle, viz.
one which has one right angle.
By prop. XII. draw two lines to meet at right angles; and when there are two given lines, make them of equal length with the other two, from the point where the former meet, Afterwards draw the inclined line. *
XXVI. To describe any proposed Triangle accu-rately.
Consider the sides of the proposed Triangle as given lines, and proceed,
When the triangle is .equilateral, according to prop. XXII.
When it is an Isosceles, according to prop. XXIIL
When it is a Scalene, according to prop. XXIV.
And when it is a right-angled Triangle, ac-cording to prop. XXV.
XXVII. To make four Lines meet at their Extre-
mities, as, for instance, those in fig. 23, pi. 2.
Let the four given lines be two of them each equal to f, and two each equal to g. Make two of them meet in a point, making any angle at pleasure, viz. make an angle and its two sides equal to the two given lines, as here the line b equal to the line g, and the line c equal to the
line/. Afterwards with the length of the other two lines describe, from the terminating points of the lines b and c which do not meet, two arcs, as here at d, and then draw two lines from the point where the two arcs intersect each other to meet the extremities of the lines b and c.
All quadrangular figures vary according as the lines are varied, and according as the angles are greater or less.
XXVIII. To draw a Quadrilateral, -cis. a Square whose sides are all equal and its angles all right ; as, for instance, fig. 20, plate 2.
Let n, fig. 17, be the given line. Draw the line ƒ equal to the line w; and proceed as in prop. XV. or using the triangular ruler, draw a line at right angles, and make this also equal in length to the line n, from the point where it meets the line ƒ. Afterwards with the line ƒ or n describe, from the terminating points of the lines/and h, where they do not meet, two arcs intersecting each other in m; and then draw the other two remaining lines. ;
XXIX. To draw a Rhombus, viz. a figure whose sides are all equal, but its' angles not right; as* for instance, fig. 21, plate 2.
Let the line k be the given line. Make two lines meet at a given or any angle, and proceed as with the preceding proposition. ..,
XXX. To draw a Rectangle or Obfong, a
Quadrilateral figure whose opposite sides are equal, and angles right angles; as, for instance, fig. 22, plate 2. .
Let ƒ and g be the given lines. Draw two lines to meet at right angles, make them of equal length with the two given ones from the point where they meet, and proceed as in prop. XXYIL
XXXI. To draw a Rhomboid, viz. a Parallelogram whose opposite sides are equal and the angles not
Let ƒ and g, again, be the given lines; and ex-cepting the right angle, instead of which another angle must be formed, the process is the same as with the rectangle in prop. XXX. or prop. XXVII.
A quadrangle, a rhombus, a rectangle, and a rhomboid, have also the common denomination of Parallelograms ; and the line which joins the two opposite angles of a four-sided figure is called a Diagonal.
XXXII. To draw a Trapezium, viz. a Quadri¬lateral figure of unequal sides, two of Which how¬ever are parallel; as, for instance, fig. 24, plate 2. .
Let/ and g (fig. 22.) be the given lines. Place the shorter line exactly on the middle of the
longer one, according to prop. VIII. Afterwards draw on that point, as at h and k, lines at right angles; make each from the greater line equal to the distance at which the short line is to be, and join the two points with a line. Then draw the two inclined lines.
Whenever kthe sides are all unequal, in any four-sided figure, such a figure is usually called a Trapezoid.
XXXIII. To draw an irregular Polygon, viz. a
Figure of more than four unequal sides ; as, for instance, the irregular Pentagon, fig. 25, plate 2. ,,
Draw the lines and angles in proper succes¬sion : viz. first draw a line of any given length, form at its extremity one of the given angles, make its yet undetermined side equal to that of another of the given lines, and form again at its extremity one of the given angles, and so on.
To copy such a drawn polygon accurately, it
must be divided into Triangles, such as those marked with dots (fig. 25), and*these Triangles must be copied in regular succession.
XXXIV. To draw a regular Polygon, viz. a
Figure of more than four sides, which, as well as the angles, are all equal; as, for instance, the Pentagon (fig. 26, plate 2).
The easiest way, when no importance is at¬tached to the sides being of more or less consider¬able length, is to describe a circle, to divide this circle into as many equal parts as there are sides required, or as the Polygon is to have angles, and afterwards to join tlwse points of division by straight lines.
A circular line, which is also called a periphery or circumference, is such only when in whatever part of this line we suppose a point, that point is every where at the same distance from a point in its middle. This latter point is called the centre.
The distance of the centre from the circumfer¬ence is called the Radius; the distance of any point in the circumference to another point in the same, in the direction through the centre, is called a Diameter; but if the direction be not to the centre, it is called a Subtense or Chord.
XXXV. To draw a circular Arc through three .
given Points which are not in a straight Line ;
• «5, for instance, through the Points I, m, n (fig.
27, plate 2).
Describe from I and m, with the same opening of the compasses, small arcs intersecting each other, as here at k and p, and likewise from the points m and« similar arcs intersecting each other, as at q and r. Afterwards draw through these points of intersection k, p and q> r two lines, and the point where these lines intersect each other is the centre of the circular arc.
•' The same process takes place when in a cir-cular arc the point, from which the arc itself has been described, is to be found, viz. by taking any three points, in the circumference, at pleasure.
XXXVI. To draw a regular Polygon upon a given Line: as>for instance, the Pentagon (fig. 26, plate 2). ' 4
Let $ t be the given line. Draw upon it, as at t, a line at right angles, making t v equal to s t, and describe with its length the circular arc s v. Then divide the length of this arc s v, into as many equal parts as the figure is to have an¬gles, here therefore into 5; take whatever poly¬gon it may be, the width, as here, from v to the point where there are still four parts remaining, and set them off from v to w, further upon the arc. Then seek in $ t and w for a point on which to rest the end of the compasses, in order to de¬scribe the curve line which passes through the points s, t, w, and set off s t four times more upon it, after having actually drawn the curve line. The rest is easy.
The width which is to be added to the divided arc consists, consequently, of the parts made, in the pentagon, of one, in the hexagon of two, in the heptagon of three parts, and so on, so that there be always four parts, left remaining.
XXXVII. To draw an Elliptical Circumference, inch, for instance, as that (fig. 28, plate 2).
Draw a line, and placing the compasses on it, describe two circles, each of which passes through the centre of the other. Afterwards with the length x a describe, upon and under that line, two equilateral triangles, and lengthen, as may piainly be seen here, the sides of these triangles; this gives the limit of the arcs required to round it, whose centres are in themngular points of the triangles above and under fig. 26.
XXXVIII. Ib divide a Line into equal parts, so that there shall be left a certain length; as, for instance, the Line b (fig. 29, plate 2,) into two equal parts, and two thirds of such a second part.
Draw any line at pleasure, somewhat longer than that which is to be divided • open the com¬passes so wide that, as there are to be two equal parts, this width may be set off twice upon the line, as is actually done here at 6. Afterwards, to get at the two thirds, divide one of the afore¬said two parts into 3 equal parts, and add two of them to the former two parts. Then, with the whole length, as has been done here, describe an equilateral triangle, take with the compasses the terminating points of the given line, and set off the length of this line, as here from c to d and from ctof Lastly, draw the line, as here d f, and intersect it by placing the ruler against the point c and against the dividing points of the
c 2
line under b; thus this line, which is equal to the given line, will be divided as required.
The same process may take place whenever a line is to be divided into a certain number of equal parts without any addition. But with re¬gard to this addition itself, it may be observed that one of the greater parts is always divided into as many equal parts as the number last men¬tioned in the addition; for instance, with three fifths into five; and that as many parts are al¬ways to be added as mentioned by the number which is first uttered, for instance with three fifths, three.
XXXIX. To make a Straight Line as long as a
Curve Line.
D|vide the curve line with dots into small parts, and set off these parts upon the straight line in the same order in which they follow each other.
The smaller the parts on the curve line, the
more accurate will be the process : however, they must not be made over small.
XL. To describe a Curve Line, of the same length
as another Curve Line.
The process is the same as in the preceding proposition.