0.5

1 cm

ON THE CURVES OF THE ATHENIAN MOULDINGS

49

If we follow the curve of the capital downwards as far as the necking Y, F A' G, is another hyperbolic

arc, of which the centre is at C, X' X' being the line in which its principal axis A' C lies. This hyperbolic

arc unites at a common tangent at the point G, a little above the neck, with the great hyperbolic arc which

forms the entasis of the columns (see Chap. V. p. 40), so that the contour of the whole column sweeps

from the ground in three hyperbolic arcs; one from the pavement to the point G, convex, another from G

to a point between F and L, concave, a third from thence to the abacus, convex. All the departures from

these bounding curves,1 at the necking, the annulets, between E and P, and above A, are mere architectural

treatments, such for instance as the flutes in a circular column, or the annulets (pd/3Soi) in the Ionic base

of the Erechtheum; and we may consider the bounding outline of the capital to be composed of a cyma

reversa produced by two hyperbolic curves. In the case of Fig. 1 (see Plate XIX.), I have subjoined

a table comparing the actual with the theoretical curve; and in the examples from other buildings I have

given sufficient measurements for any one so disposed to make the trial with the hyperbolic elements

which I have determined.'2

One point, in which these Athenian examples of the Doric echinus differ from the more usual form,

is that the abacus projects a little over the echinus. The length (C A) of this projection when obtained

accurately (by averaging a sufficient number of examples) is found to be exactly that of the principal

semi-axis of the hyperbola forming the echinus of the capital; and I think that we may feel satisfied that

it was so left by these scientific architects as a testimony of perhaps one of the earliest solutions of the

properties of the conic sections, namely the determination of the centre of the curve.'1

Fig. 2, which is placed near the former, is an example from one of the fallen capitals from the

southern peristyle. The width of the abacus in these, as I have already remarked (Chap. II. p. 15),

is considerably less than in the eastern or northern capitals. The appearance of the echinus of some of

these on the south side suggests the idea that part, at any rate, of the amount of this reduction was made

after some or all of the capitals were worked ; for (as in this example) the more convex part of the

echinus, as shown by the dotted line, is cut off, in order apparently to reduce the spread of the capital.

This would have affected its entire breadth about .036, no insignificant matter in the eye of a Greek.

Fig. 3 is from a fallen capital of the Pronaos ; this is a segment of an hyperbola, and is identical

in its curvature with Fig. 1, except that the same curve (the curve APL in Fig. 1) is continued

throughout the whole length of the surface, viz. from A to E ; consequently the lower part of this echinus

is a flatter curve than that of the peristyles.

The projecting points of the annulets lie here in the curve which forms the Epitrachelium, and these

two curves do not die into each other, as in the case of the peristyle.

Fig. 4 is from the column of the larger order of the Propylaea. This example is also hyperbolic,

and we find the centre C here also pointed out by the face of the abacus B O produced. The principal

Appendix, Art. 5) is equal to | ; and it is remarkable that in all the

hyperbolic arcs connected with these mouldings which I have investigated,

this ratio has always a simple manageable value. Thus, in the echinus of

the Parthenon and of the Theseum 5:1 = 3:2. In the epitrachelium of

the Parthenon (the curve G A' F in Fig. 1), e: 1 = 3 :1. In the echinus

of the Propylsea both in the large and smalt orders e: 1 - */ 2 : 1, a ratio

easily constructed geometrically. In the soffit of the pediments of the

Parthenon e - '\. In that of the Propyls e = 2. Now (although I do

not deny that these curves may possibly have been drawn by a mechanical

apparatus) since all these ratios are easily constructed, it follows that by the

well known method of focus and directrix (see Appendix, Art. 5), the curve

might have been easily laid down by points.

1 The late Mr. Jopling, who bestowed very great attention on the

subject of curved lines in architecture, and whose discoveries in the con-

struction of mechanical curves deserve to be extensively known, contends,

in a letter read to the Institute of British Architects, Feb. 21, 1848, that

the contour of the echinus of the columns of the Parthenon is produced by

one of the forms of the conchoid of Nicomedes (see Appendix, Art 3), a

curve of great variety and beauty, and which has the advantage of being of

somewhat easier construction than the hyperbola. If we were bound to

find a single curve to fit the moulding from A to E (instead of touching the

point L), a conchoid might probably be found which would approximate

more nearly than any single hyperbola which could be drawn between these

points. But still the coincidence of the conchoid would be by no means

complete. Moreover the echinus of the columns of the Pronaos (Fig. 3) is

a simple hyperbola along its whole section from A to E, and there are

several other cases of the use of the hyperbola, such of the curve G A' F

(Fig. 1), which cannot be conchoidal. These proofs, together with the

evidences of selection which I have already pointed out in the elements of

the curves, and the exceeding accuracy of the coincidences, so far as I claim

them, are sufficient proofs that the curves in question are hyperbolic. Yet it is

not unlikely that other examples, especially in the forms of vases, may be found,

which may justify the belief that the Athenian architects were acquainted

with the conchoid. It is true that this curve is mentioned by the writers

on the geometry of the ancients, as having been discovered about two

centuries later than the period of which we are speaking; but the same

authorities also attribute the invention of the conic sections to the school

of Plato, of which curves some instances can be shown to have been used

in the time of Pericles, and indeed much earlier.

- The profiles given in these plates were obtained by the aid of

Professor Willis's cymagraph, an instrument somewhat resembling the

pantograph. In the instances here given, the process has been in every

case reversed in the same instrument, in order to eliminate the small errors,

to which it occasionally is liable in certain parts of its operation.—These

errors, however it should be stated are so extremely slight as to be un-

important in ordinary cases. The larger curves, as Pigs. 10, n, etc., were

taken partly by the cymagraph and partly by other methods.

3 I have already observed, in a note in Chap. V. p. 40, that there is

reason to believe that the curve we are here speaking of was used in the

earliest Greek temples. But as one of its branches might have been drawn

by means of the focus and directrix, without it being at all necessary that

the centre should be known, it is very interesting to find this proof of

progress in the science of mathematics recorded in the works of the

architect. Comp. Appendix, Art, 5.

49

If we follow the curve of the capital downwards as far as the necking Y, F A' G, is another hyperbolic

arc, of which the centre is at C, X' X' being the line in which its principal axis A' C lies. This hyperbolic

arc unites at a common tangent at the point G, a little above the neck, with the great hyperbolic arc which

forms the entasis of the columns (see Chap. V. p. 40), so that the contour of the whole column sweeps

from the ground in three hyperbolic arcs; one from the pavement to the point G, convex, another from G

to a point between F and L, concave, a third from thence to the abacus, convex. All the departures from

these bounding curves,1 at the necking, the annulets, between E and P, and above A, are mere architectural

treatments, such for instance as the flutes in a circular column, or the annulets (pd/3Soi) in the Ionic base

of the Erechtheum; and we may consider the bounding outline of the capital to be composed of a cyma

reversa produced by two hyperbolic curves. In the case of Fig. 1 (see Plate XIX.), I have subjoined

a table comparing the actual with the theoretical curve; and in the examples from other buildings I have

given sufficient measurements for any one so disposed to make the trial with the hyperbolic elements

which I have determined.'2

One point, in which these Athenian examples of the Doric echinus differ from the more usual form,

is that the abacus projects a little over the echinus. The length (C A) of this projection when obtained

accurately (by averaging a sufficient number of examples) is found to be exactly that of the principal

semi-axis of the hyperbola forming the echinus of the capital; and I think that we may feel satisfied that

it was so left by these scientific architects as a testimony of perhaps one of the earliest solutions of the

properties of the conic sections, namely the determination of the centre of the curve.'1

Fig. 2, which is placed near the former, is an example from one of the fallen capitals from the

southern peristyle. The width of the abacus in these, as I have already remarked (Chap. II. p. 15),

is considerably less than in the eastern or northern capitals. The appearance of the echinus of some of

these on the south side suggests the idea that part, at any rate, of the amount of this reduction was made

after some or all of the capitals were worked ; for (as in this example) the more convex part of the

echinus, as shown by the dotted line, is cut off, in order apparently to reduce the spread of the capital.

This would have affected its entire breadth about .036, no insignificant matter in the eye of a Greek.

Fig. 3 is from a fallen capital of the Pronaos ; this is a segment of an hyperbola, and is identical

in its curvature with Fig. 1, except that the same curve (the curve APL in Fig. 1) is continued

throughout the whole length of the surface, viz. from A to E ; consequently the lower part of this echinus

is a flatter curve than that of the peristyles.

The projecting points of the annulets lie here in the curve which forms the Epitrachelium, and these

two curves do not die into each other, as in the case of the peristyle.

Fig. 4 is from the column of the larger order of the Propylaea. This example is also hyperbolic,

and we find the centre C here also pointed out by the face of the abacus B O produced. The principal

Appendix, Art. 5) is equal to | ; and it is remarkable that in all the

hyperbolic arcs connected with these mouldings which I have investigated,

this ratio has always a simple manageable value. Thus, in the echinus of

the Parthenon and of the Theseum 5:1 = 3:2. In the epitrachelium of

the Parthenon (the curve G A' F in Fig. 1), e: 1 = 3 :1. In the echinus

of the Propylsea both in the large and smalt orders e: 1 - */ 2 : 1, a ratio

easily constructed geometrically. In the soffit of the pediments of the

Parthenon e - '\. In that of the Propyls e = 2. Now (although I do

not deny that these curves may possibly have been drawn by a mechanical

apparatus) since all these ratios are easily constructed, it follows that by the

well known method of focus and directrix (see Appendix, Art. 5), the curve

might have been easily laid down by points.

1 The late Mr. Jopling, who bestowed very great attention on the

subject of curved lines in architecture, and whose discoveries in the con-

struction of mechanical curves deserve to be extensively known, contends,

in a letter read to the Institute of British Architects, Feb. 21, 1848, that

the contour of the echinus of the columns of the Parthenon is produced by

one of the forms of the conchoid of Nicomedes (see Appendix, Art 3), a

curve of great variety and beauty, and which has the advantage of being of

somewhat easier construction than the hyperbola. If we were bound to

find a single curve to fit the moulding from A to E (instead of touching the

point L), a conchoid might probably be found which would approximate

more nearly than any single hyperbola which could be drawn between these

points. But still the coincidence of the conchoid would be by no means

complete. Moreover the echinus of the columns of the Pronaos (Fig. 3) is

a simple hyperbola along its whole section from A to E, and there are

several other cases of the use of the hyperbola, such of the curve G A' F

(Fig. 1), which cannot be conchoidal. These proofs, together with the

evidences of selection which I have already pointed out in the elements of

the curves, and the exceeding accuracy of the coincidences, so far as I claim

them, are sufficient proofs that the curves in question are hyperbolic. Yet it is

not unlikely that other examples, especially in the forms of vases, may be found,

which may justify the belief that the Athenian architects were acquainted

with the conchoid. It is true that this curve is mentioned by the writers

on the geometry of the ancients, as having been discovered about two

centuries later than the period of which we are speaking; but the same

authorities also attribute the invention of the conic sections to the school

of Plato, of which curves some instances can be shown to have been used

in the time of Pericles, and indeed much earlier.

- The profiles given in these plates were obtained by the aid of

Professor Willis's cymagraph, an instrument somewhat resembling the

pantograph. In the instances here given, the process has been in every

case reversed in the same instrument, in order to eliminate the small errors,

to which it occasionally is liable in certain parts of its operation.—These

errors, however it should be stated are so extremely slight as to be un-

important in ordinary cases. The larger curves, as Pigs. 10, n, etc., were

taken partly by the cymagraph and partly by other methods.

3 I have already observed, in a note in Chap. V. p. 40, that there is

reason to believe that the curve we are here speaking of was used in the

earliest Greek temples. But as one of its branches might have been drawn

by means of the focus and directrix, without it being at all necessary that

the centre should be known, it is very interesting to find this proof of

progress in the science of mathematics recorded in the works of the

architect. Comp. Appendix, Art, 5.