One other question may be considered here. Does a series ever occur in which three units are repeated regularly, instead of one or two? In experiments we discovered that the subject found it impossible to feel repetitions of three in a series, and the only way that such a series was tolerable was when the three could be grouped somehow into one or two units. Therefore we should not expect to find such repetitions frequently, if at all.

To sum up: Do series always end with a heavier unit? Are units equally distant from each other more adapted to continuous or run-on railings, while units with symmetrical arrangements within themselves are found more often where separateness of objects enclosed is more aimed at than their connection? Is a less heavy end found after symmetrical series than after the other kind? Are repetitions of three units used at all, and if so in what way?

Obviously the only illustrations of these questions will be found in the arrangement of posts and pillars in balustrades of whatever description. In these cases alone do we find repeated series, with repetitions within the unit, as well as of the unit as a whole. The following examples have been taken by looking over about one thousand photographs and by recording every instance that occurred.

Having 100 illustrations of repetitions of groups, with units repeated equidistantly between them, and of elements distinctly symmetrical, several new factors came to light. In all the one thousand photographs looked over, not a single instance was found of unit-groups with the units within, arranged at other than equal distances. There were many variations in the number of units in the groups; but the number being given, the units were arranged at equal distances from each other wherever the effect desired was of detached sections or of continued series. There are obvious structural reasons for this. Any repetition of groups for a balustrade or protective railing, which is the almost exclusive use of this variety of repetition, would be weakened by wider apertures on either side of the centre. A reasonably enclosed space is necessary to make the railing of value, therefore the specifically symmetrical unit as opposed to the rhythmic unit was found always in carvings, scrolls, bas-reliefs, etc., alternating with vertical supports. We should expect, then, in general, that in railings where an aspect of continuity of progress along some border or a tendency to go around an enclosure was sought, the units would be rhythmic in character, impelling one to motion and to carrying the eye and general organism out of repose into movement. We should expect, on the contrary, that symmetrical units would be found where repose or partial distinctness of the separate elements enclosed was desired, and where the attention was not to be carried away in so marked a degree. Seventy-three of the one hundred illustrations were of balustrades where the rhythmic factor was presumably aimed at.

The Rathaus at Braunschweig had a symmetrical design alternately occurring, but with four in a section, so that the section as a whole was not symmetrical and the attention was driven on, and in the other cases some other effect than rhythm was obviously aimed at. The genius of the structures was heavy and massive and the balustrade made in keeping with them, since an effect of motion or rhythm would have clashed with the spirit of the whole.

These examples have all been of the balustrades around enclosures, balconies, etc. Since the rhythmic unit has been found more fitting for them, we should expect, conversely, that in front of separate unities, such as windows, doors, etc., the symmetrical unit would be more in evidence. At first sight, the facts do not seem to bear us out in this. Of twenty-seven examples of separate windows, doors, and gates enclosed by railings, only four had distinctly symmetrical designs. (Casa Palladio, Bergamo Chapel, Petit Trianon.) These are wrought-iron designs in the centre with repeated rods on each side, or a row of six pillars with the central two larger and more decorated. Twenty-three, however, remain to be accounted for, and the solution of the difficulty is observed at once in the distinction between odd and even numbers. As was previously suggested there are obvious difficulties in having posts in a balustrade at any but equal distances, since the gaps left by unequal distances from the centre would destroy their reason for being. This difficulty can easily be overcome in wrought iron by extra central decoration, although it is not always done by any means; but in stone balustrades, unless there is carved open-work, or solid reliefs, there is no other choice than repeated posts, either divided into sections or continuous, and no variation is possible except to have an even or odd number of them. We should then expect that there would be an odd number in separate detached enclosures, bringing a post in the centre to emphasize the balance, while in a continuous series each group would have an even number, thus giving no centre to fixate upon, but driving the attention on without repose at any one point more than another. It might seem doubtful that any such refinement should have been actually expressed in architecture, but examination of these examples shows this treatment to be very general. Of the twenty-three examples of separate enclosed details, eleven have an odd number of posts. Of the ten that remain, four are examples of windows along the side of a building, with separate detachments of balustrade in front of each. By having an even number of group-units the continuity of the row is maintained in spite of a separation of the sections. Two of the ten are sections of balustrade over the central doorway of a building. These balustrades are divided into three sections, of which the centre is widest and the ends only half as wide. Thus, although there are six posts in the central section, the balustrade as a whole is distinctly divided into a bilateral symmetrical arrangement. Three of the others have an even number of pillars, but they support an odd number of arches; and the arch, not the pillar, is taken as the unit of the repeated series. (Arches will be discussed later.) The one example unaccounted for represents a number of possible cases, where for some reason, following out a general scheme of building, or what not, the odd number is not insisted upon for separate clusters. But the fact that only one out of twenty-three is thus unexplained shows an unmistakeable tendency in the other direction.

A distinction between odd and even numbers cannot be felt above eight repetitions without actual counting, and often not even then.

The two final exceptions are of a gate and a decoration over a door (Fontainebleau, Piacenza) where there are nine or more units in the group. It is impossible to feel the system of this arrangement, and the result is proportionately confusing. A reservation must be made here concerning iron railings. There is no discrimination between odd and even in the number of iron rods in a section of railing and no tendency to symmetrical designs rather than rhythmic before detached enclosures. This is because from the nature of the case, there is no distinction possible between odd and even in the number of slender iron rods necessary to enclose a space with any security. There must of necessity be so many of them that the difference cannot be perceived, and so slight is the importance of each rod that the effect is more of a variegated surface than of actual beats of a rhythm. As soon as iron is wrought into large enough shapes, each repeated detail is of the same importance as in stone, but the slender rods commonly used in iron railings, although their repetition is rhythmic like all the others, give too slight a motor impulse to carry the attention past the heavy limits of whatever they enclose. They are found in front of many windows, but on account of the lightness of their rhythm compared with the solidity of limiting piers, no confusion results.