An introduction to linear drawing

Front matter

an

INTRODUCTION

to

Introduction and Preface

INTRODUCTION.

An elementary treatise on Drawing, adapted to the use of common schools, can not but be well received. Besides the professions which make the art of drawing their particular study, anatomists, naturalists, mechanicks, travellers, and indeed all persons of taste and genius, have need of it, to enable them to express their ideas with precision, and make them intelligible to others.

Notwithstanding the great utility of this branch of education, it is a lamentable fact, that it is seldom or never taught in the publick schools, although a very large proportion of our children have no other education than these schools afford. Even in the private schools where drawing is taught, it is too generally the case that no regard is paid to.the geometrical principles on which the art depends. The translator appeals to experience when he asserts, that not one in fifty of those who have gone through a course of instruction in drawing, can do more than copy such drawings as are placed before them. Being ignorant of the certain rules of the art, (and they are the most certain because mathematical) they are always in leading strings, and, unless endowed with uncommon genius, never originate any design, and rarely attempt to draw from nature. It is to remedy this defective mode of teaching, that the translator has been induced to present this little work on the elements of drawing, to the American publick.

Mot of our faculties, when exercised, may attain to a surprising degree of perfection. A precision may be acquired by the eye and hand, almost equal to that of ordinary instruments. With this view, the society for the improvement of elementary instruction in France, directed some of their most distinguished members to procure a work on the art of drawing, which should be applied to the system of mutual instruction, there the national system. The following treatise is, in a great measure, a translation of that approved by them. It is not intended for a treatise on the art in all its numerous branches, but merely the linear, and, of course, the fundamental and most useful part of it.

The geometrical figures are arranged according to the difficulty of their execution, rather than in the order of theorems.

Each figure is accompanied with suitable explanations, so that the teacher or monitor will easily comprehend them, and be able to teach them to his classes, without much previous acquaintance with the art.

The pupils are each furnished with a slate and pencil. The monitor directs what figure shall be drawn, and if the pupils are not all furnished with this treatise, he chalks the figure on a board, painted black for the purpose, and suspended where all can see it. The slates are then examined by the monitor, and precedence is given to whichever pupil has executed the figure best.

The instructer should select a sufficient number of the most skilful for monitors, who should be under his immediate instruction. As soon as they have become expert in drawing the figures of the first class, a second may be formed to be instructed by the first class, (which now becomes the second) and so on to the sixth. The highest class under the master, may consist of about fifteen pupils. The lower classes may consist of any number, but for every six or eight scholars there should be a monitor.

The first class draw right lines, angles, parallels, perpendiculars and triangles.

The second class draw polygons, and polyedrons, or solid figures of many sides.

The third class make circles, and regular polygons.

The fourth draw a protracter, make angles of a given opening, draw ellipses, cylinders, cones, spheres, &c.

The fifth apply the preceding figures to architectural drawings, vases, and tasteful ornaments.

The sixth class draw the orders of architecture, and such other objects as an ingenious instructer shall direct.

If the school do not consist of more than 30 or 40 pupils, there will be no need of employing as monitors any but the highest class.

The children should not be permitted to draw on paper, until they have become thoroughly acquainted with the figures of the five first classes. Before they attempt the sixth, they may be permitted to review the five preceding classes, drawing the figures on paper with a lead pencil.

The pupils are not to be allowed the use of a rule, or any other instrument; but the monitor, to correct and prove their figures, may be furnished with a rule, dividers, square, and protracter or graduated semicircle. The rule should be a good one, with the inches and tenths of inches marked on it, that, when the pupils have become expert in making the figure, the difficulty may be increased by requiring the whole, or some part of the figure, to be of a given length or dimension.

On most rules in common use, the inches are divided into quarters and eighths, but as it is our plan to apply Geometry to decimal arithmetick, such rules as are divided into tenths should be preferred. When the simplicity of decimal calculations is so evident, it is to be regretted that all our measures are not subdivided into decimal parts, as our currency is, and why our government should set so good an example in one particular, and neglect all he rest, it is not easy to determine.

Although this treatise was originally designed for schools of mutual instruction, still a slight examination of it will show that there is nothing which unfits it for use in schools on any other plan. If the pupils are all taught, and their drawings examined by the instructer, they will do well; but if they are likewise required to examine and correct each other's work, they will do better; they will acquire a familiarity with the figures, and an exactness in execution, to which mere learners seldom attain.

PREFACE

TO

THE SECOND EDITION.

The favourable reception of the first edition of this Treatise, has induced the Translator to revise it carefully, and to add to it a Second Part, containing the elements of Perspective Drawing, to which the First Part is a good introduction.

Questions, also, upon the more important parts of the book are added; and the Translator hopes that this more correct and enlarged edition will meet with the same favour that a liberal publick has bestowed upon its predecessor.

Dec. 1827.

First class - Right lines

PART FIRST.

FIRST CLASS.

The first class only draw right lines, triangles and perpendiculars. The corrections are made with a rule, and dividers.

The four first figures drawn above, relate particularly to the first eight propositions. To ascertain if the line be straight, let the monitor draw a line through it or near it with a rule. To ascertain if a line be cut into equal parts, measure the parts with the dividers, if the eye be not sufficiently practised to detect the errors without their assistance.

PROPOSITIONS.

1. Draw a right line (that is, a straight line.) fig. 1.

2. Draw a right line and divide it in the centre. fig. 2.

3. Draw a right line and cut it into quarters. fig. 3.

4. Draw a right line and lengthen it a much farther. fig. 2.

5. Draw a right line and continue it twice length. (fig. 4.)

6. Draw a right line and lengthen it three times its length. (fig. 3)

7. Cut a right line into three equal parts. (fig 4.)

8. Cut a right line into six or eight equal parts, and so on.

It will be a useful exercise at this stage of the business, to show the parts of a line when divided. Thus, if required to show how much three quarters of a line are, the pupil must find one quarter, and the rest of the line will be three quarters. To find three fifths of line, cut the line into five parts, and take three of them. A very correct idea of fractions may in this way be communicated.

9. Draw a line one inch long, then two, three, four, five, six. (fig. 5.)

10. Draw a line and divide it into inches.

It is of no consequence what the length of the line is. Begin at the left and mark as many whole inches as there may be.

11. Draw a horizontal line.

A horizontal line is one drawn from left to right, or from right to left. The surface or top of a bowl of water is horizontal or level.

12. Draw a perpendicular line. (fig. 6.)

A perpendicular or vertical line is one perfectly upright, as a string, with a weight at the end of it, will hang from a nail, or from the hand.

In making horizontal lines, the pupil should make them parallel to the top or bottom of his slate or paper, and in making perpendiculars they should be parallel to the sides of the slate or paper. Parallel lines are lines running in the same direction equally distant from each other in every part; thus, the horizontal lines in figures 1, 2, 3, 4, are parallel to each other. Lines may be drawn parallel at any distance from each other.

18. Draw two parallel horizontal lines, then three, four, five and six.

14. Draw two parallel perpendiculars, then three, four, five and six.

15. Draw an oblique line and cut it into two, four, three and six parts.

An oblique line is one between a horizontal and a perpendicular; that is, a leaning line.

16. Draw two parallel obliques, then three, four, five and six.

17. Draw parallel lines an inch apart, then half an inch, a quarter, &c.

18. Draw a perpendicular, and cut it into two, three four, five, six equal parts.

It is difficult to cut a perpendicular into equal parts, because of an optical deception which leads us to think the upper parts shorter than they really are. This must be guarded against.

19. Join two dots or points by a right line.

The pupil will move his pencil two or three times from the left dot to the right, before he draws the line. This precaution is more necessary when the operation is performed on paper than when on a slate, where it may be erased if wrong.

20. Make an acute angle. (fig. 7.)

Care must be taken to distinguish an angle, from what is called its point or apex. The angle is the opening between two lines that meet, and the point or apex is the point where the lines meet. A pair of dividers forms a number of different angles, by being opened more or less.

It is this opening of the sides which determines the size of the angle, and not the length of the sides, which, if lengthened out ever so far, would not affect the size of the angle, because the opening will only be the same part of a a great circle that it was of a small one.

Imagine two lines which cross each other as in figures 9 and 10. They will make four angles. These are right angles if they are equal, and they will be equal if one line is perpendicular to the line it crosses. If the angle be less than a right angle, it is called an acute angle; if more, it is called an obtuse angle. Acute means sharp, and obtuse means blunt.

21. Make an obtuse angle. (fig. 8.)

22. Make an acute angle with the opening turned upward, downward, to the right and to the left.

23. Make a triangle. (fig. 11.)

Close the space between the sides of an angle with a right line, and you make a triangle, a figure which has three angles and three sides.

The base is the side on which the triangle is supposed to rest.

The apex of a triangle is the point opposite to the base.

The height of a triangle is a perpendicular drawn from the apex to the base. In the figure it is shown by the dotted line.

A triangle is called Isoceles when two sides are equal. If all three of the sides are equal, it is Equilateral, (which word means equal-sided;) and if all the sides are unequal, it is called Scalene.

24. Raise a perpendicular on a horizontal. (fig. 9.)

This will produce right angles, as we have before remarked. To ascertain if the angle be exact, take a piece of what is called bonnet paper or thin pasteboard, cut it round and then cut the round piece into quarters. Each quarter will have two sides at right angles, and by inserting the apex into the opening of the angle drawn by the pupil, any incorrectness will be detected. A small brass or iron square will serve the same purpose, but does not satisfactorily show that a right angle is equal to a quarter of a circle, which is also called a quadrant.

25. Cross a right line with a perpendicular. (fig. 10.)

The right line should be drawn in various directions, to show the pupil that a perpendicular may be raised on any right line, whether horizontal or oblique.

26. Draw a rectangular, or right angled triangle, (figs. 12 and 13.)

This is a triangle of which one of the angles is a right angle, as the lower left hand one in fig. 12, and the top one in fig. 13. The base may be horizontal or inclined.

27. Make a rectangular isoceles triangle.

There is no difference between this and figures 12 and 13, except that in an isoceles triangle, two of the sides must be of equal length. In fact, fig. 12 is an isoceles.

Figure 13, though rectangular, is a scalene also.

28. Draw a rectangle. (fig. 14.)

A rectangle is properly a figure with four sides, of which each two opposite sides are equal and parallel, and of which all the angles are right angles.

The lower side is the base, and the right or left side is the height.

To ascertain its correctness, the Monitor may examine every angle with his quadrant of pasteboard, or he may with his dividers see if the left hand upper, and right hand lower angles are as far apart as the other two angles are. Figure 14 is what is often called a long or oblong square.

29.Make a rectangle, and cut it into equal right angles. fig. 15.

30. Make a parallelogram, and mark its height. (fig. 16.)

The parallelogram, like the rectangle, has its opposite sides parallel, but none of its angles are right. The height is a perpendicular dropped from the top to the base, and is marked by the dotted line in the figure.

31. Make a square. (fig. 17.) This figure has all its sides equal, and all its angles right.

32. Draw two angles with parallel sides. (figs. 18 and 19.)

Two angles, as in figure 18, are called parallel, not because their sides are of equal length, but because their openings and points correspond exactly. Fig. 19 is designed to exercise the pupil in making parallel angles in various positions.

33. Draw obliques equidistant (that is, equally distant) from a perpendicular.

Draw first a horizontal, raise a perpendicular on its centre, and then draw a line from the top of the perpendicular to each end of the horizontal. The figure will then be an isoceles triangle, as in fig. 11.

34. Make a scalene triangle. (fig. 13.)

As it is not difficult to make a triangle of unequal sides, it will be well for the monitor to prescribe the length of one or more of them. Thus he may say: "Make a scalene triangle, of which the three sides shall measure one inch and two tenths; one inch; and eight tenths of an inch."

35. Make an equilateral triangle. (fig. 20.)

After the pupil makes the figures exactly, let the length of the sides he given, as one, two, three, &c. inches. Then require the point to be under the base, turned to the right, &c.

36. From a given point draw a perpendicular.

First draw a right line, then make the proposed point, and lastly draw the perpendicular.

27. Raise a perpendicular on the end of a right line.

38. Make a Rhomb or Lozenge. (fig. 21.)

The four sides are equal as in the square, but the angles are not right angles. To draw this figure, make a right line, cross it with a perpendicular, like the dotted lines in the figure, and then draw the sides.

If the Rhomb or Lozenge have all the angles equal, the figure is merely a square placed obliquely, as in fig. 22.

39. Cut a rectangle into halves. (fig. 23.)

This will make two angles, whose exactness may be tested by an eighth part of a circle of pasteboard, the rectangle being quarter of a circle, as was stated under Prop. 24.

40. Cut an acute angle into two equal parts. (fig.24.)

41. Double an angle.

Make an angle of any size, and then make another of the same size by the side of it. Suppose the lower angle of fig. 24 to be made first, then by making the upper right line, the angle will be doubled.

42. Triple an angle. (fig. 25.)

43. Cut an angle into three equal parts. (fig. 25.)

44. Cut an angle into six equal parts. (fig. 26.)

These three propositions need no explanation.

Second class - Solids, Figures, &c

SECOND CLASS.

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1. Make two triangles of perpendicular sides.

After having made one angle, the pupil will draw a perpendicular to one of the sides, and then a perpendicular to the other side, until the perpendiculars cross each other.

One of the angles is acute, and the other obtuse; and if you lengthen one of the perpendiculars, a new angle will be formed exactly like the angle first made, as the dotted continuation of the perpendicular in fig. 1 shows.

2. Make two triangles of perpendicular sides. (fig.2.)

Make one triangle, and then draw a perpendicular to each side, until the perpendiculars touch and form angles. Each side of each triangle must be perpendicular to some side of the other triangle.

3. Make a trapezoid. (fig. 3.)

A trapezoid has four sides, of which, two, called the bases, are parallel. In the figure, these are the upper and lower sides. The height is a perpendicular from base to base. As this figure is easily made, the length of the bases and the height may be given: thus, "Make a trapezoid whose height shall be one inch,and whose bases shall be an inch and a half and two inches." When no two of the sides are parallel the figure is called a Trapezium.

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4. Make a six sided polygon of unequal sides. (fig.4.)

The word polygon means many-angled. To make a polygon, the best method is, first to place dots at the angles and then draw right lines from dot to dot.

5. Make a five sided polygon of unequal sides. (fig.5.)

6. Make two polygons of unequal but parallel sides. (figs. 5 and 6.)

7. Make a six sided polygon of equal sides.

8. Make a five sided polygon of equal sides.

A polygon of equal sides is called a regular polygon.

9. From one point of a polygon draw diagonals, and then draw a parallel polygon within the first. (fig. 7.)

After drawing a polygon, from either of the points carry right lines to all the rest, thus making several triangles. These lines are called diagonals. Then you have only to draw parallels to the several sides from diagonal to diagonal.

To vary the exercise, let the pupil draw a polygon outside of the first drawn. He will then only have to lengthen the diagonals.

10. Draw a polygon, and from a central point draw diagonals, then draw a parallel polygon within and outside of the first drawn. (fig. 8.)

11. Make a triangular pyramid. (fig. 9.)

First draw the triangle which forms the base, then place a dot for the point or apex, and draw right lines from the point to each angle of the base.

12. Draw a quadrangular (or four angled) pyramid. (fig. 10.) then cut it by a plane parallel to its base.

The process is the same as in the preceding figure. The plane, or parallel to the base, must be the last thing done. The height of the pyramid is a perpendicular dropped from the apex or summit to the base. The pupil must be careful to distinguish the front lines from the back lines of the figure.

13. Make a six sided pyramid. (fig. 11.)

14. Make a five sided pyramid. (fig. 12.)

15. Make a five sided pyramid, and cut it by a plane parallel to the base. (fig. 12.) Note. When the base of the pyramid is a regular polygon, and the height falls upon the centre of it, the pyramid is upright and regular; such are figures 10, 11 and 12.

16. On two polygons of parallel sides, raise a pyramid, (fig. 12.)

This is merely another way of performing the last figure, by drawing both polygons before you draw the trunk of the pyramid.

17. Make an upright triangular prism. (fig. 13.)

A prism is a body formed from two equal and parallel polygons, whose corresponding points are joined by lines, all parallel, and equal to each other; such are figures 13 to 20 inclusive.

The height of a prism is a perpendicular to the two bases. A prism is said to be upright when its sides are perpendicular, and oblique when its sides lean or are inclined.

18. Make an oblique triangular prism. (fig. 14.)

19. Make an oblique five sided prism. (fig. 15.)

20. Make an upright parallelopiped. (fig. 16.)

When the bases of the prism are parallelograms, the body is called a parallelopiped. All the six faces are then parallelograms, and the two opposite faces are equal.

21. Make an oblique parallelopiped. (fig. 17.)

22. Make a Cube. (fig. 18.)

The cube is a parallelopiped, all of whose faces are equal squares, and each placed at right angles with the contiguous or next squares.

A cube is a solid square, but it will be perceived, that in consequence of perspective, only the front and back faces appear square. These two faces should be traced first, and the rest will easily be added. Dice are cubes.

23. Draw an oblique cube. (fig. 17. if you make its sides equal.)

24. Cut a prism by a plane parallel to its bases. (fig. 19.)

Third class - Circles, &c

A Plane is a level. Place one die upon another, and together they make a prism, cut by a plane where the separation between the dice is.

25. Make a six sided prim, and double it by

lengthening it. (fig. 19.)

When one die is put upon another, the first die is doubled in length.

26. Make a jive sided prism, and cut it by three planes parallel to its base.

A long prism may be drawn and cut as in Prop. 24, or a short prism be first made and lengthened as many times as you please.

27. Draw a Jive sided prism in a horizontal position. (fig. 20.) Cut it by a plane parallel to its base.

THIRD CLASS.

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1. Describe a circle and mark its centre, (fig. 1.)

By constant practice the pupil will be able to draw a circle, and mark its centre with great exactness. The monitor with a pair of dividers will prove it. The pu- pils in the Monitorial School have various methods of marking circles, without the aid of dividers, the most (8)

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14, Describe a circle and draw a tangent to it.

Tangent comes from a latin word, which means to touch. A tangent is a right line which touches a cir- cle, but does not cut any of it off. If a right line be drawn from the centre of the circle or arc to the point of contact (that is, the point where the tangent touches the circle) the two right lines, (that is, the radius and the tangent) will be perpendicular to each other, (fig.9.)

The monitor may test this with his quadrant of pasteboard ; or, marking two places on the tangent at equal distances from the point (f contact, he may see with his dividers if these points are at equal distances from the centre of the circle.

15. Draw four tangents to a circle, forming a quadrilateral or four sided figure.

This figure need not form a perfect square, as in figure 10.

16. Circumscribe or surround a circle with a square.

(fig- 8.)

(fig. 10.) When the four tangents make right angles with each other, the figure is a square. In other cases, any di- rection may be given to two of the tangents.

In figure 10 we say the circle is circumscribed by the square, or the circle inscribed in a square*

17. Inscribe a square in a circle. (fig. 11.)

When a polygon has all the points of its angles touch- ing a circle, it is said to be inscribed in a circle, and the circle circumscribes the polygon.

18. Double an arc of a circle. (fig. 6.)

This is more difficult than Prop. 11. First draw an arc and mark the centre of its circle, then prolong the arc to two, three, Stc. times its former size.

19. Draw a tangent to a circle from a given point outside. (fig. 8.)

20. Draw two tangents to a circle from a given point. (fig. 12.)

Observe that in drawing a tangent to a circle in problem 14, any part of the circle may be taken, but when a tangent is drawn from a given point, it can hit but two points of the circle, as in fig. 12.

21. Cut a circle into six equal parts, or, in other words, inscribe a regular hexagon in a circle.) (fig. 13.)

The radius of any circle is equal to one side of the hexagon to be inscribed in it. The monitor, therefore,

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(13) may measure the radius with his dividers, and then ap- ply them to each side of the hexagon. In other words, the cord of an arc, which is the sixth part of a circle, is equal to a radius or half diameter, (usually called a semi-diameter.)

22. Cut a circle into three equal parts, and inscribe an equilateral triangle, (fig. 13.)

After the hexagon is correctly drawn by problem 21st, it is easy to inscribe the triangle required in this, by drawing a cord between two points of the hexagon.

If cords be then drawn between the three remaining points of the hexagon, another triangle will be formed, whose base will be opposite the base of the other trian- gle, forming a beautiful figure resembling a star.

23. Make two unequal circles tangent outside.

nequal circles are circles of unequal size only.

24. Make two unequal circles tangent inside, (fig.15.)

25. The centres and the point of contact being giv- en, perform problems 23 and 24.

The monitor will mark the centres, he. When the circles touch either within or without, the point of con- tact and the two centres will be in a right line, and these may be tested with a rule.

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14.) A regular polygon has all its sides equal, and all its angles of an equal opening. When such a polygon is inscribed in a circle, the sides are cords of equal arcs, and the points cut the circle into equal parts.

26. Inscribe a regular octagon in a circle. (fig. 16.)

Draw two diameters perpendicular to each other, then divide each quarter of the circle into halves by other diameters ; then draw arcs from diameter to di- ameter.

27. Inscribe a regular pentagon in a circle. (fig. 17.)

It is difficult by the eye alone to divide the circum- ference into five equal parts, and the object of this problem is to exercise the pupils.

28. Make a triangle, and circumscribe a circle.

First make a triangle, and then the object is to de- scribe a circle which shall cut each of its three points. To do this, raise a perpendicular on the middle of one

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(fig. 18.) of the sides, and then on the middle of another side. These perpendiculars will cross each other, and the point of section (that is, the point where they cut each other) will be the centre of the circle required.

In the figure, the dotted lines show the perpendiculars and centre.

29. Make a circle, and draw a tangent triangle. (fig. 19.)

Three tangents to a circle are easily made, but the monitor may increase the difficulty by giving directions to the tangent sides. Thus, let two sides be at right angles, obtuse or acute; let the triangle be equilateral, &c.

30. Draw a regular pentagon, and circumscribe it with a circle. (fig. 17.)

31. Draw a regular hexagon, and circumscribe it with a circle. (fig. 13.)

32. Draw a regular octagon, and circumscribe it with a circle. (fig. 16.)

In the former problems, the circle was made first, now the polygon.

33. Inscribe a circle in a triangle. (fig. 19.)

To find the centre of the circle, draw a line from the middle of either side of the triangle to the apex opposite, then do the same by another side and its opposite apex; the place where these two lines cross each other, will be the centre of the circle to be inscribed. See the dotted lines in fig. 19.

34. Make an arc which shall pass through two given points. (fig. 20.)

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After having marked two points, trace an arc of a circle which shall pass through both of them. The centre must be somewhere on a perpendicular to the middle of a cord which would join these two points.