CIVIL ARCHITECTURE. 



605 



Practice, is to consist of quadrants, (which, in this example, are 

 '"Y""*' 12,) at the points 1, 2, 3, &e. Draw CE, at any angle 

 with AB, equal to two of these parts : draw 11 f per- 

 pendicular to CD, equal to CE : Anvrfg perpendicu- 

 lar to II f, equal to 11 Q: draw gk perpendicular to 

 gf, equal to 10 P. Proceed in this manner until all the 

 sides of the spiral fret are drawn. Produce liyto 1, 

 fg to 2, gk to 3, and so on. Upon the centre at 11, 

 with the distance d B, describe the arc B 1 ; from the 

 centre f, with the distance/" 1, describe the arc =1, 2 ; 

 from g, with the distance g 2, describe the arc 2, 3. 

 Proceed in the same manner with all the remaining arcs, 

 the last terminating at 12. Then, from the centre r, 

 with the radius r 12, describe a circle for the eye of the 

 volute ; and the radius, r 12, of this circle will be equal 

 to AD as required, provided the operation be accurately 

 performed. 



After the same manner may the inner spiral be drawn. 

 Fig. 1. No. 2. shows the construction of the centres to a 

 larger scale, and by a method not so liable to inaccuracy, 

 as follows : Having made <//"equal to CE, divide it into 

 halves at e ; draw e r perpendicular to df, equal to e d 

 or ef, and draw dinr, and fk or. Makees, st, tr t 

 equal each to G 2 on the scale : draw it If, nto parallel 

 to df; mzkefg perpendicular to df, equal to 1 1 Q : 

 draw the diagonal g Ip parallel to d i n r : draw k I, op, 

 n m, i h, perpendicular to df: draw g h parallel to 1 If; 

 draw the diagonal hmq: draw Im and pq parallel to 

 fd, and d,f, g, k, a, c, will be the centres. This me- 

 thod is not so liable to error, as taking the parts suc- 

 cemively from the scale, and applying them to the spi- 

 ral fret in the same order. 



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=rf l=CB+j= the radius of the first 

 quadrant, and d A=/1=CB x the radius of the se- 

 cond quadrant; therefore, df=d l/"l=(CB+i) 

 (CB j;) = 2j:, the first side of the fret ; and CE will 

 also be equal to twice x, as by the construction ; likewise, 

 let nbe equal to the number of parts in CD ; then will 1 F, 



the first parallel distance on the scale, be equal to ; 



tt 





the third 3H= ; but 



and the second 2 G= 

 n 



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 



term 



is , and the last term =2*; therefore, (x-f-f ] 



X"=nx-|-T=DC-f-4:, is the sum of the series; but 

 DC + xis=CA+j r = CB + o: r, equal to the length 

 of the spiral fret. Now, let this last quantity be taken 

 from CB-f- 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 

 passing through their centres, and the junction of their 

 arcs. 



The methods hitherto discovered for the description 

 of volutes, are 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 1'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, are very unequal, as the distances of the ad- 

 joining revolutions in the two upper quadrants are 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 resource, 

 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 has been long 

 known, it is singular that its application to the Ionic vo- 

 lute has never been hinted 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 formed 

 by the progression of the correponding sides of a series 

 of similar triangles, as shown in the second volume of 

 that work. 



Fig. 2. No. 1. To describe the Ionic volute similar 

 to that in the temple of Erectheus, by the principles of 

 the logarithmic spiral, the centre O, the cathetus OA, 

 and the distance AI, 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 GO A 

 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 O 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 ap- 

 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 are to be drawn as the spiral 

 abed efg h i. 



Another method of doing this, without marking the 

 drawing, is to make a scale AB, with the proportional 

 compass in the same manner as marking the distances 

 from the centre, viz. having set the proportional corn- 



Practice. 



PLATE 

 CLXXXtV. 

 Fig. 2. 

 No. 1. 



To de- 

 scribe (lie 

 \nolute. 



Another 

 method. 

 Fig. 2. 

 No. 2. 



