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PRACTICAL GEOMETRY.
PLATE I.
FIG. 1.
IN a given square, A B C D, to describe a regular octagon. With one half the distance A C or B D, and on A B C D, as centres, describe the arcs g m d, e m k, c m h, and f m i; join f e, d c, k i, and h g g, and it will be the octagon required.
FIG. 2
In a given circle to describe an equilateral triangle. Upon any point, A, in the circumference, with the radius A G, describe the arc B G F; draw B F, and make B D equal to B F; join D F and D B, then B D F will be the equilateral triangle required.
FIG. 3
In a given circle to describe a square or an octagon. Draw the diameter A C and B D at right angles; join A B, B C, C D and D A, and A B C D will be the square. For the octagon, bisect A B in E, and carry A E eight times around the circumference.
FIG. 4, 5 and 6.
Upon a given line, a b, to construct any regular polygon. Upon a and b as centres, with a radius equal to a b, describe two arcs in tersecting each other at f; from b draw the perpendicular b c, and divide the arc a c into as many equal parts as the polygon is to have sides. Through the second division d draw b g; make e f equal to f d, and through e draw a g, meeting b g at g; then g will be the centre, and g a the radius of a circle, that will contain a b to any number of sides required.
FIG. 7.
To describe the segment of a circle by means of a triangle. Let A B be the length of the segment, and C D the perpendicular height in the middle; through the points D and A draw D A, and draw D E parallel to A B; make D E equal to D A, and join E A, which makes the triangle E D A. Put in pins at the points A D B ; then move your triangle around the points D and A, and the angular point will describe half the segment. The other half will be described in the same manner, which will complete the whole segment required.
FIG. 8.
The transverse and conjugate axes A B and C D of an ellipsis being given, to find the two foci, and from thence to describe an ellipsis. Take the semitransverse A E or E B, and from C as a centre describe an arc cutting A B at F and G, which will be the foci.
Fix pins in those points, a string being extended about the points F C G, then move the point C around the fixed points F and G, keeping the string tight, and the ellipsis will be described.
FIG. 9.
To describe an ellipsis by an instrument called a trammel. Set the distance of the first pin, B, from the pencil at C, to half the shortest diameter, and the distance of the second pin, A, to half the longest diameter; the pins being put in the grooves, as shown in the figure; then move the pencil at C, and it will describe the ellipsis required.
CHAPTER 2MOULDINGS.
PLATE II.
FIG. 1, 2 and 3.
To describe the Grecian echinus, or ovolo to any given height and projection. Figures 1, 2 and 3 are all of one height, but of different projections. Divide their heights (excluding the quick) and their projections, each into a like number of equal parts; through the divisions 1 2 3 and 4, on the line of height, draw lines parallel to the moulding, and also from A, at the extremity of the outline, draw lines to 1 2 3 and 4, on the line of projection; then through the points, where these lines cut those first described, trace the outline of the echinus, and continue it as judgment directs from A around the quick to the fillet. The outline just described is a parabola, and that of fig. 4 is an hyperbola, which may be drawn by leaving off the lower fourth part, or more or less of the line of height A D; or, which produces the same thing, from the points on the line of height at C, draw lines which, if produced, would meet in a point at a distance of once or twice the projection of the moulding, or more or less, remembering that the nearer the moulding these lines meet, the nearer will the outline of the echinus approach to a straight line.
It will be plainly seen, that, by the application of the conic sections either to the echinus cimarecta or cimareversa, an endless variety of beautiful outlines may be obtained.
To describe the cimareversa, fig. 5, divide A B and 3 D each into a like number of equal parts, into six for instance, as here, and draw A 1, A 2, A 3, and C 1, C 2, C 3; then trace the curve as before directed. To describe a scotia, fig. 6, join the ends of each fillet by the right line A B; bisect A B at D; through D draw C D F parallel to the fillets, and C D and D F, each equal to the depth of the scotia; divide D A, D B, A G and B F, each into the same number of equal parts; from E, and through 1 2 3 on A G, and also through 1 2 3 on B F, draw lines. Then, from the point C, and through the points 1 2 3 on the line A B, draw lines, cutting the former lines at 1 2 3 4 5 6, and trace the curve through those points.
CHAPTER 3TUSCAN ORDER.
PLATE III.
THE Tuscan order is nearly the same in its proportions as the Doric. Of this order we have no complete example remaining from antique buildings. There is one given by Vitruvius, from whom we derive all we know concerning the origin of the order, of which the column is seven diameters high, including base and capital, and diminishes one fourth part of its lower diameter. But the cornice projects more than one fourth part of the height of the column, which renders the example unfit for our practice.
In the example here given, I have, in imitation of Vitruvius, made the column seven diameters high, including base and capital, and the entablature two diameters; although in practice I do not always follow this rule, since, in different situations, both column and entablature require different proportions. They should in all cases be proportioned according to the weight, or apparent weight, which they have to sustain. It will sometimes be convenient to lower the entablature, in which case it may be reduced to one hundred and ten minutes, or any where between one hundred and ten and one hundred and twenty, taken in about equal proportions from the frieze and architrave, and leaving the cornice its full proportions. The base given in this example may, when expense is to be avoided, be kept off with propriety; and particularly when the column is seven diameters or less. The mouldings in the capital and cornice are Grecian, although the order is of Roman origin.
To draw the Tuscan order to any given height, the height given must be divided into nine equal parts, because, the column being seven and the entablature two diameters, one ninth of the height is the diameter of the column just above the base. Suppose it be required to find the diameter of the column to a height of twenty-one feet. One ninth of twenty-one feet is two feet four inches. Of that length make a scale of sixty equal parts, which are called minutes. First divide two feet four inches into six equal parts, like the scale a b; then divide a 10 into two equal parts, and a 5 into five parts. Each of the last divisions will be one minute, or a sixtieth part of the diameter of the column.
Under H and against each member of the order is figured the number of minutes in their respective heights, and under P, the number of minutes which each member projects. The measures of the order are to be taken from the scale of minutes above, as will be seen by the following directions given for drawing the base. Against the plinth c d is fifteen minutes; take fifteen minutes from your scale, and set it down on a vertical line drawn for the side of the column. Against the torus is twelve minutes; set that down on the same line; and against the fillet e f is three minutes; set that down also. Under P and against the plinth at g is ten minutes, the projection of the plinth; set that down at right angles with the column, and draw the line g across the plinth. Against the torus at h is ten minutes, its projection; with six minutes in your compasses describe the half circle, which completes the torus. Against the fillet at i is four minutes, which set off for its projection; then, with four and a half minutes in your compasses, describe the part of a circle which joins the fillet to the column, and the base is completed. Go on in the same way to describe the remaining part of the order.
A is a section of the cornice showing the bedmould recessed up into the soffit of the corona.
B is a section of the band to the architrave.
C is the capital drawn on a large scale for the purpose of showing the exact curve of the ovolo.
CHAPTER 4GRECIAN DORIC ORDER.
PLATE IV.
FIG. 1.
THE district of Argolis first received colonies who introduced civilization into Greece. It has been reckoned the cradle of the Greeks, the theatre of the events which produced their earliest annals, and the country which produced their first heroes and artists. It was in the temple of Juno at Argos, where the Doric order first arose to a marked eminence, and became the model for the magnificent edifices afterwards erected in the other cities, states, and islands. After the Doric order had been established in the temple of Juno at Argos, it was employed in the temple of Jupiter Nemeus, between Argos and Corinth; Jupiter Olympius, at Olympia in Elis; in a splendid triple portico in the city of Elis; and also in three temples in the same city, to Juno, Minerva, and Dindymene; at Eleusis, in the great temple to Ceres; in that of Minerva, at Suniurn; in the temple of Minerva Parthenon; in the temple of Theseus; in the entrance to the Acropolis, and other public edifices, of great magnitude and splendor, at Athens. In many of the islands there were also temples of the Doric model; that of Apollo in the isle of Delos; Juno in Samos; Jupiter Panellenius, Ægina, and Silenus, in Sicily; and innumerable in places of inferior note. Even in Ionia, it was employed in the temple of Apollo Panionius. Many of these edifices were of great magnitude. The temples of the Greeks were universally of an oblong form; in some, the porticos were at the ends only; in others, they were extended quite around the cell, some in single, others in double ranges; some were covered with roofs, others were left partly uncovered; and some of them were divided by ranges of pillars along the middle of the cell. The superstructure was placed upon a platform composed of three steps, which surrounded the whole edifice, and upon which the columns were all placed without bases. The number of columns was either six along the ends and thirteen along the sides, or eight along the ends and seventeen along the sides, counting the angular columns twice. When formed upon so large a scale and the ranges of columns so distinctly insulated, and when artists of the first talents not only formed the models of the edifice, and directed its execution, but also, under the strongest influence of rivalship and thirst of glory, with their own hands clothed those edifices with sculptures, and enriched them with statues, under these circumstances the essential parts of the Doric order produced effects not to be exceeded for simplicity and majesty.
That the proportion which the different members of the Doric order bore to each other was practised by the Greeks with considerable latitude, and that the ancients did not scrupulously adhere to any precise proportion, will be evident from the following table, which exhibits the dimensions of the principal parts of this order in many of the ancient Greek edifices, which have been examined with accuracy. The dimensions here put down are in diameters and minutes.
In every Greek Doric, the vertical face of the architrave projects beyond the inferior diameter, but is within the superior one. In the temple of Corinth, and the Doric portico at Athens, the ovolo or echinus is of an elliptical shape; but in every other instance of Greek capitals, it is hyperbolical, excepting the single instance of the portico of Philip, king of Macedon, which is a straight line.
The number of annulets in the capital varies from three to five; and the number of horizontal grooves which separate the shaft from the capital varies from one to three.
Elevation of the Doric Order on the Temple of Theseus at Athens.
This temple is one of the most ancient and beautiful examples of that order now existing. It was erected about ten years after the battle of Salamis, by Cimon, the son of Miltiades. The ceiling of the porch is remarkable for its construction, having large beams of marble with the upper sides level with the bed of the cornice, and the ends corresponding exactly to the triglyphs in the frieze, which gives the idea of the disposition of the timbers first used in buildings, and from which the Doric order had its origin. This building is adorned with beautiful sculpture. The metopes of the frieze are charged with historical figures, in which are represented various exploits of Theseus.
The height of the column is five diameters and forty-two and a half minutes, and that of the entablatures two diameters and ten minutes, so that column and entablatures, both together, are seven diameters and fifty-two and a half minutes. If the column was seven and a half minutes higher than it is, its diameter would be one eighth part of the entire height; and it will be near enough in practice to divide the whole height into eight parts; then, giving one part to the diameter of the column, proceed to draw it as before directed in the Tuscan order. The column diminishes thirteen minutes, and its sides are composed of straight lines or those gently curved.
Fig. 2 is a plan of the soffit inverted, showing the distribution of the drops, which are two and a half minutes in diameter, and one and a half in length.
Fig. 3 is a section of the cymatium on a large scale.
Fig. 4 shows the capital on a large scale, in which the cantor of the ovolo, and also the shape of the annulets, are more plainly seen.
CHAPTER 5ROMAN DORIC ORDER.
Doric Order, as approved by Sir William Chambers.
"Vignola's Doric being composed in a greater style, and in a manner more characteristic of the order, than any other, I have chosen it for my model; though, in particular members, I have not scrupled to vary, when observation taught me they might be improved.
"Perfect proportion, in architecture, if considered only with regard to the relations between the different objects in the composition, and as it merely relates to the pleasure of the sight, seems to consist in this; that those parts which are either principal or essential, should be contrived to catch the eye successively, from the most considerable to the least, according to their degrees of importance in the composition, and impress their images on the mind, before it is affected by any of the subservient members; yet, that these should be so conditioned, as not to be entirely absorbed, but be capable of raising distinct ideas likewise, and such as may be adequate to the purposes for which these parts are designed.
"The different figures and situations of the parts may, in some degree, contribute to this effect; for simple forms will operate more speedily than those that are complicated, and such as project will be sooner perceived than such as are more retired; but dimension seems to be the predominant quality, or that which acts most powerfully on the sense; and this, it is apprehended, can only be discovered by experience; at least to any degree of accuracy. When, therefore, a number of parts, arranged in a particular manner, and under particular dimensions, excite, in the generality of judicious spectators, a pleasing sensation, it will be prudent, on every occasion where the same circumstances subsist, to observe exactly the same arrangement and proportions; notwithstanding they may in themselves appear irregular, and unconnected.
"In composing the orders and other decorations which are contained in the present publication, this method has constantly been observed; the author having himself, with that view, measured with the utmost accuracy, and not without some danger, many ancient and modern celebrated buildings, both at Rome and in other parts of Europe; strictly copying such things as appeared to be perfect, and carefully correcting others, which seemed in any degree faulty, relying therein not alone on his own judgment, in doubtful cases, but much on the opinion and advice of several learned, ingenious artists of different nations, with whom he had the advantage of being intimately connected when abroad.
Excerpted from The Architect, or, Practical House Carpenter (1830) by Asher Benjamin. Copyright © 2014 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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The superbly illustrated and detailed handbook that popularized the use of classic Greek architectural style in America in the early and middle 1800s. 271 illustrations.