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divide CE and D F into the same number of equal parts. Through the points 1, 2, 3, &c. in CD, draw the lines 1 a, 2 b, 3 c, &c. parallel to AB; also through the points 1, 2, 3, in CE and DF, draw the lines 1 A, 2 A, 3 A, cutting the parallel lines at the points a, b,c; then the points a, b, c, are in the curve of the parabola.
Prob. 33. To describe an hyperbola.
If B and C are two fixed points, and a rule A B be made moveable about the point B, a string ADC being tied to the other end of the ruler, and to the point C; and if the point A be moved round the centre B, towards G, the angle D of the string A DC, by keeping it always tight and close to the edge of the rule A B, will describe a curve D HG, called an hyperbola.
If the end of the ruler at B were made moveable about the point C, the string being tied from the end of the ruler A to B, and a curve being described after the same manner, is called an opposite hyperbola.
The foci are the two points B and C, about which the ruler and string revolves.
The transverse axis is the line IH, terminated by the two curves passing through the foci, if continued.
The centre is the point M, in the middle of the transverse axis I H.
The conjugate axis is the line N O, passing through the centre M, and terminated by a circle from H, whose radius is MC, at N and O.
A diameter is any line V W, drawn through the centre M, and terminated by the opposite curves.
A conjugate diameter to another, is a line drawn through the centre, parallel to a tangent with
either of the curves, at the extreme of the other diameter terminated by the curves.
Abscissá is when any diameter is continued within the curve, terminated by a double ordinate and the curve, then the part within is called the abscissa.
Double ordinate is a line drawn through any diameter parallel to its conjugate, and terminated by the curve.
Parameter, or latus rectum, is a line drawn thirough the focus, perpendicular to the transverse axis, and terminated by the curve.
Prob. 34. To describe an hyperbola by finding points in the curve, having the diameter or axis A B, its abscissa B G, and double ordinate D C.
Through G draw E F, parallel to C D; from C and D draw CE and DF, parallel to BG, cutting E F in E and F. Divide C B and B D, each into any number of equal parts, as four; through the points of division, 1, 2, 3, draw lines to A. Likewise divide EC and D F into the same num
of equal parts, viz. four ; from the divisions on C E and D F, drảw lines to G; a curve being drawn through the intersections at G, a, b, &c. will be the hyperbola required.
Perspective is the art of drawing upon a plane surface the outlines of objects, so as to give the same representations to the eye that the objects themselves do in nature.
This art is of the utmost importance in drawing, and its study cannot be dispensed with by those
who wish to make any proficiency in this art. Some knowledge of it ought to be acquired, previous to the study of every branch of drawing, whether that of the figure, landscape, flowers, &c.; for though its utility may not appear equally evident in all these, yet there are many cases in each, where it is of indispensable necessity; and an acquaintance with it will save the student the trouble of much wrong thinking, and will enable him to avoid many errors which he otherwise must necessarily fall into.
It is true, that many people learn to draw without studying perspective, and too many of those whose profession it is to teach this useful branch of education, do not sufficiently recommend to their pupils to learn the principles of this science. The reason of this it is not difficult to point out. The study of perspective, like that of geometry, has in itself but few charms, and it is only by being well convinced, that without it we can never hope to arrive at excellence, and by experiencing how much it accelerates and assists us in our practice of drawing, that we can with patience and resolution go through studies that certainly appear to most people dry and unentertaining. Unfortunately, from this circumstance, and from the abstruse and obscure manner in which it is treated off by the generality of writers on the subject, few will take the trouble of making themselves acquainted with a science so necessary.
But though the understanding perspective thoroughly, certainly does require considerable geometrical knowledge, added to great patience and persevering investigation, yet so much as would enable those who draw to avoid making any very
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glaring blunders, and render their study of drawing much more pleasant and easy, is so far from being difficult, that it is astonishing any one should hesitate a moment about acquiring it.
Some part of the blame may, however, be fairly laid to the want of a treatise on perspective, of an easy and popular kind, such as might suit those who have had no opportunities of acquiring a knowledge of geometry. It would be in vain to endeavour to supply this deficiency properly in a work of so limited a nature, and which embraces so many subjects as the present; yet though our plan will not permit us to treat of it so fully as it deserves, we shall lay down, as concisely as possible, a few of its principal rules, the understanding of which will be found useful to beginners in the art of drawing. Explanation of the principal Terms used in
Perspective. The perspective plane is the surface of the picture itself, which may be imagined to be a plane of glass placed upright between the spectator and the objects to be drawn. Then, if lines or rays be supposed to come from every part of the objects to the spectator's eye, when viewing them through the glass, they would cut the plane in certain points; and if these points were connected by lines, they would give the perspective representation. It is upon this simple idea that all the rules of perspective are founded : they are so many methods of finding out the above-mentioned points; and when the light, shadow, and colour, are added, the whole constitutes a picture exactly resembling the original.
Visual rays are the rays of light which come from the different parts of the objects to our eyes.
Point of sight is the spectator's eye. This has been erroneously confounded with the centre of the picture, as will be seen afterwards.
What is called an original line, is any line in Nature, or in the objects themselves, which are to be drawn in perspective.
An original plane is any surface or plane of the objects to be represented.
If we suppose a line to proceed from the eye, parallel to any line in the objects we are viewing, and to continue till it arrive at the picture or perspective plane, the point where it would touch the plane is called the vanishing point of the line.
All lines that are in Nature parallel to each other have the same vanishing points. The reason of this will be easily seen, if the reader considers the method of finding the vanishing point of an original line just mentioned; for the same line which would find the vanishing point of one, will do for them all, and form only one point.
If we could suppose a plane or surface to proceed from the eye of the spectator, in a direction parallel to any side of an object which we view through the plane of glass, and if it continue till it arrive at the glass, the line which it would form by contact with the glass, is called a vanishing line ; and it is the vanishing line of the side of the object which the supposed plane was parallel to.
In every picture or perspective plane, there is a point, where a line drawn from the eye perpendicular to the picture, would touch it; this point is called the centre of the picture, and is the same which is often called in old books on perspective, the point of sight. But this is a wrong term for it;