In order to show the truth of this universal method, we shall subjoin the following demonstration, which, it is hoped, will be satisfactory to the reader.
Let x represent any of the equal parts of which DC consists, and r the radius of the eye.
Then will d B=d 1=CBA-x= the radius of the first quadrant, and d A=f I=CB—x the radius of the se cond quadrant ; therefore, d f=d 1—f=(CB- x (CB—x)=2x, the first side of the fret ; and CE will also be equal to twice x, as by the construction; likewise, let n be equal to the number of parts in CD ; then will IF, the first parallel distance on the scale, be equal 4 x and the second 2 G=—; the third 3 —; but if the half sum of the extreme terms of this arithmetical progression be multiplied by their number, the product will be equal to the sum of the series. Now, the first .
x 2 x and the last term =2 x; therefore,( x±x=DCA-s, is the sum of the series ; but DCA-x is=CA-Fx—r=_CB-I-x--r, equal to the length of the spiral fret. Now, let this last quantity be taken from CB-}-x, the radius of the greatest quadrant, and there will remain r the radius of the eye as ought to be ; for the difference of the radii of any two adjoining qua drants is equal to the side of the spiral fret between their centres, or, in other words, in the straight line ing through their centres, and the junction of their arcs.
The methods hitherto discovered for the description of volutes, arc extremely imperfect and limited to three revolutions, and the eye to a certain portion of the height of the spiral. The method shown by Palladio, said, by Sir William Chambers, to be that of De l'Orme's, is imperfect, as the two adjoining arcs of any two revolu tions cannot have the same tangent at their junction ; and the straight line, which is a tangent to any two arcs, can only have one point of contact in the same revolu tion: but though this imperfection is remedied in Gold man's, yet, in the latter, the successive distances, on the lines of junction between two adjoining revolutions of the spiral, arc very unequal, as the distances of the ad joining revolutions in the two upper quadrants arc nearly equal, whereas, in the two lower quadrants, the distan ces between the adjoining revolutions of the same spiral decrease very unequally towards the centre ; and this ir regularity is extremely unpleasant to the eye.
But though the method we have shown is the most perfect and easy of any yet published, and has also the advantage of being universal in its application, it is far from being perfect, as it cannot be applied where the volute consists of many spirals, as in the Ionic order of the temple of Erectheus at Athens, and much less any of the other two which preceded it. There is one re
source, however, by which this evil can be remedied, and by which perfection alone can be obtained, and this by the principles of the logarithmic spiral ; for, in all other methods, the fillets and the intervals of the volute are never in continued geometrical proportion.
Though the principles of this curve have been long known, it is singular that its application to the Ionic vo lute has never been hintod at by any author, before the publication of the Principles of Architecture by Mr Pe ter Nicholson.
The following method of describing the Ionic volute, upon the principles of the logarithmic spiral, by means of a proportional compass, is not so liable to error, and is much more expeditious in practice, than a scale form ed by the progression of the corresponding sides of a series of similar triangles, as shown in the second vo lume of that work.
Fig. 2. No. 1. To describe the Ionic volute similar to that in the temple of Erecthcus, by the principles of the logarithmic spiral, the centre 0, the cathctus 0 A, and the distance A I, between the first and second revo lutions of the outer spiral, being given.
Produce AO to E, and draw GOC at right angles to AOE ; bisect the angles AOC and COE by the right lines BOF and DOH, and the angles EOG and GOA will also be bisected. Find a mean proportional between OA and OI, and make OE equal to it ; find also a mean proportional between OA and OE,and make OC equal to this mean; likewise find a mean proportional between OA and OC, and make OB equal to this last mean. Set the proportional compass, so that the distances between the two pair of points will have the same ratio to each other as OA to OB; then opening the distance between the widest pair of points from 0 to A, and turning the com pass to the narrowest pair, make OB equal to that dis tance. Again, taking OB with the widest end, make OC equal to the distance of the narrowest end. Pro ceed in this manner, by alternately taking the length of the last radius found between the widest points, and ar plying the distance between the narrowest points from the centre upon the succeeding line as a radius, until the last radius approach the eye, then through all the points draw a curve, which will be one of the spirals. In the same manner, and at the same setting of the compass, each of the inner spirals arc to be drawn as the spiral abcdefghi.