VOLUTE, one of the principal ornaments in the Ionic capital composed of two or more spirals of the same species, having one common eye and centre, variously channelled, or hollowed out in the form of mouldings.
Among the remains of ancient architecture, the Ionic capitals of the temples of Ereehtheus and Minerva Pans, at Athens, arc the most beautiful. The spiral best adapted for this purpose, is known by the name of the logarithmic spiral; the method of describing which is as follows : Draw a straight line for the cathetus; take any point in the straight line as a centre; through this point draw an other straight line at right angles, and these two straight lines cutting each other, will form four right angles : bisect any two adjacent right angles, and let the bisecting lines be produced on the other side of the centre, and the whole will be divided into eight equal angles by as many lines, upon which the radii are to be placed.
In the calculation we are here about to make, that part of the eat hetus above the centre is supposed to be 20 minutes at a medium, and the next radius 18.3; both these being given, the calculation is founded upon the following principle : Let a = 20, and 6 = 1S.3, 6' then a : L :: L : which gives the third radius.
a 7, which gives the fourth radius. a a a 62 63 6 y which gives the fifth radius.
Therefore, any radius may be found, independent of the rest : thus, x being a variable quantity, representing " the numbers of the radius. Thus, suppose the eighth radius 6: were required ; make x = 8, then — = — 2 The arithmetical operation will be best performed by loga rithms : now, as 6 = 18.3, and a = 20, multiply the loga rithm of 18.3 by '7, and the logarithm of 20 by 6, subtract the latter product from the former, and the remainder will be the logarithm of the answer.
Now the logarithm of 18.3 is 1.26245, which, multiplied by 7, gives 8.83715 : and the logarithm of 20 is 1.30103; which, multiplied by 6, gives 7.80618 : then 8.83715 7.80618 = 1.03097, which is the eighth radius.
The following method will serve to prove the result of the operations, which are all dependent on each other ; and will save the trouble of frequent reference to the Logarithmic Tables.
Log. 20 = 1.301030 And thus, having gone through one revolution, the radii of the remaining revolutions may be found in a similar 4 manner.
Now to apply the number thus found : Plate I.—The centre is marked by the point 0; upon the cathetus set 20 from 0 upwards, which will give the extremity of the first radius; upon the second radius towards the left, set 18.3 from the centre, which will give another point in the curve; then, following round in the same progression, from the centre, set 16.74, 15.32, 14.01, 12.82. 11.73, 10.73, 9.82, &c. upon each suceeeding radius respectively, to 2.37, and three points will be ti und in each quadrant.
In the Plate here referred to, we have only retained one decimal place, the scale not admitting of any more. And note, when the second decimal is 5, or more, the first decimal is made one more.
In the first quadrant, take the length of the middle radius, viz. 18.3: set one foot of the compasses in 20, then describe an arc near to the centre; with the same radius set the foot of the compasses in 16.7, and describe an are cutting the former ; then, from the point of intersection, as a centre, describe an arc through all the three points 20, 18.3, 16.7. Proceed in the sante manner with every quadrant, by taking the middle radius; and from the extremity of each outer radius, describe two arcs intersecting each other; and from the point of intersection, as a centre, with the same radius describe an arc through the two extremities, until you arrive at 2.4 ; then with radius 2.4 describe a circle for the eye, and the whole spiral will be completed.
Should it be required to describe a spiral through any given point, so as to form a fillet, divide the distance from the centre to that point into twenty equal parts, form a new scale, and proceed in the manner already described.