Louis-Benjamin Francœur

“ 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. ”

“ To draw a house which stands oblique to the picture, that is, one which has no side parallel to the picture or Perspective plane, begin with the base line C D and the horizontal line A B. Take Ethe corner of the building nearest to the spectator's eye, and draw the line E B for the bottom of one side of the building.
Then, to find the Vanishing point of the lines of the other side, from the Centre of the picture, which you may fix at F, draw the perpendicular F G equal to the Distance of the picture. ”

“ 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. ”

“ 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. ”

“ Now to give each figure its relative proportion ac- cording to its distance, draw vanishing lines to the van- ishing point from the extremities of the nearest figure, and in the angle thus formed will be found the heights of all the other figures. To find the height of a figure on any part of the road, only lead off a horizontal line, parallel to the base line, for the foot of the object, and the height is found by raising a perpendicular from the horizontal until it strikes the other converging or vanishing line. ”

“ 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. ”

“ 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 ”

“ 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. ”

“ 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.
Figs 7-8 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. ”

“ These arcs on a solid globe would be parallel to the equator, but do not appear so on a plane or flat surface. In geography, they are called Parallels of Latitude.
If an apple be taken and sliced from side to side, it will exactly represent the circles, which are planes cutting a sphere perpendicular to its axis. ”

“ If you please, you may take half the base or half the height, before you multiply, and then there will be no need of halving the result. A triangle is always the half of a parallelogram of the same base and height.
The pupil, it is to be hoped, need not be told that 12 inches make a foot, and 3 feet or 36 inches an English yard. We advert to this, because we have hitherto only measured by inches, and it may be well to say that when feet or yards, inches and decimals are named together, the yards or feet must be brought into inches. ”

“ When is a polygon said to be inscribed in a circle ? 15.
When a tangent is drawn from a given point, how many points of a circle can it touch ?
16.
What is the proportion between the radius of a Circle and one side of a hexagon inscribed in it ?
17.
What is meant by unequal circles ? (p. 20.)
18.
What is meant by a regular Polygon ? (p. 21.)
19.
What is a regular hexagon ?
20.
What is a regular pentagon ?
21.
What is a regular octagon?
22.
How will you circumscribe a triangle with a circle ?
23.
How will you find the Centre of a circle to be in- scribed in a triangle ? ”

“ 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. ”

“ 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. ”

“ Not that there exist in fact, exact and rigorous proportions and rules which are never deviated from, for art has not those fixed rules which are found in the sciences. Certain proportions having been ordinarily observed, and by the consent of all persons of good taste, being found the most suitable, these proportions should be considered as a rule not to be deviated from 5 without good reasons. The draughtsman, by strictly observing these proportions, is secured from criticism, is sure of doing well, and of obtaining the approbation of judges. ”

“ Or—if the circumference of the base of a column be measured with a string, and multiplied by 0,159, the module will be found ; and from this, the height of the whole edifice, and of all its parts.
Pediments are triangular structures, whose height may be much varied according to their extent. There are some whose height is a third, fourth, fifth or sixth of the base. This proportion is left to the taste of the artist ”

“ A radius drawn from the centre to the number 30 on the graduated semicircle, will form an angle of 30 degrees with the diameter of the semicircle. And so for any other number of degrees. It will be seen that any number of degrees less than 90 will make an acute angle, and more than 90 degrees will form an obtuse angle thus, in fig. 4, 30 degrees form an acute angle, and the remaining 150 degrees of the half circle form an obtuse angle. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ To draw the Chimney, find its actual height above M, the actual height of the gable end, and continue the perpendicular D E M to O. Draw the line O B, then, oneach side of the centre P L, lay off on the base line two spaces Pa and Pb each equal to half the real breadth of the chimney, and raise the perpendiculars ac and bd.
To find the thickness of the chimney, lead off a line from O to Q equal to its real thickness. Draw the line Q B. Lead a horizontal from c till It strikes the line Q B at e ”

“ How many bricks will it take to construct a wall, 300 feet long, 6 feet high, and 1,5 thick ?
Note. It will be seen that this and the preceding cal- culation make no allowance for mortar.
6. A well is 6,9 yards deep, and 1,2 yards in diam- eter. I wish to make a wall in it, 0,4 yards thick ; how many cubick feet of stone will build it ?
Calculate the well as if it were to be entirely filled up. Its diameter being 1,2, its radius must be half that, or 0,6 tenths ; to this add, 0,4 tenths, the proposed thickness of the wall, and you have 10 tenths, or a whole yard, for the radius of the well. ”

“ 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. ”

“ Artists and mechanicks ought to be able themselves to measure their work, whatever it may be; and to draw plans, to make contracts for work, to calculate the price and quantity of materials necessary for the work ”

“ 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. ”

“ This will lead the pupil to conclude, that objects of the same size appear the smaller the farther they are from the eye, and the larger the nearer they are to it, though the real size of the object is unchanged. ”

“ After thousandths, come ten thousandths, hundred thousandths, mil- Konths, &c. but for ordinary uses we seldom come down to so small a fraction. Multiplication is performed as if there were no comma ; but in the sum total, as many figures must be cut off at the right hand of the comma, as are cut off in both the multiplier and sum multiplied. ”

“ In drawing a pyramid, must the front or back lines of the base be uppermost ?
. 17. What is a quadrangular pyramid ?
18.
When a pyramid is to be cut by a plane, what part must be drawn last ?
19.
What is the height of a pyramid? ”

“ I have measured round a basin, and find the dis- tance 28,5 inches, and conclude by Problem III, Sect. 1, that the radius is 4,53. The rest of the work is done like the above example. Ans. 64,49 sq. in.
4.
The circumference of a circle is found to be 4,85 feet, what is its superficial contents ? Ans, 1,76 sq. ft.
Problem VI. To find the surface of an upright cylinder. (4th Class, fig. 12.)
Rule. Multiply the circumference of the base by the height. As the base is a circle, knowing the radius, it is easy to find the circumference. ”

“ A man wishes to cover the walls of a room with cloth. The length of all the sides added together is 675 inches, 7 tenths, the height is 98,4 the cloth is 32 inches wide ; how long a piece will cover the walls ? ”

“ If 1,994 thousandths be the radius or half diameter, 3,988 will be the whole diameter of the column. Six figures are separated, because there are three in the multiplier, and 3 in the multiplicand. The three right hand decimals are unimportant. ”

“ 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. ”

“ From the same angles draw two lines, one to the right and the other to the left of the vanishing point, and at equal distances on each side of it. Then draw a line to connect the two points where these two lines inter- sect the vanishing lines, and the square is finished. ”

“ Example 1. A painter has painted a circular hall, the walls are 3,4 yards high, and the diameter of the hall is 54,2 yards, how many square yards has he painted ? Ans. 579,156 sq. yds.
If there be doors and windows, they are calculated separately, and subtracted from the amount. To find the contents of mouldings, measure them with a piece of string or parchment, which will yield to their various curvatures, and if you please, add their width and sur- face to the first amount. ”

“ Figs 12-13 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. ”