Витрувий De Architectura libri decem

Marcus Vitruvius Pollio:
de Architectura, Book III
Introduction
1. The Delphic Apollo, by the answer of his priestess, declared Socrates the wisest of men. Of him it is said he sagaciously observed that it had been well if men''s breasts were open, and, as it were, with windows in them, so that every one might be acquainted with their sentiments. Would to God they had been so formed. We might then not only find out the virtues and vices of persons with facility, but being also enabled to obtain ocular knowledge of the science they profess, we might judge of their skill with certainty; whereby those who are really clever and learned would be held in proper esteem. But as nature has not formed us after this fashion, the talents of many men lie concealed within them, and this renders it so difficult to lay down an accurate theory of any art. However an artist may promise to exert his talents, if he have not either plenty of money, or a good connexion from his situation in life; or if he be not gifted with a good address or considerable eloquence, his study and application will go but little way to persuade persons that he is a competent artist.

2. We find a corroboration of this by reference to the ancient Sculptors and Painters, among whom, those who obtained the greatest fame and applause are still living in the remembrance of posterity; such, for instance, as Myron, Polyclitus, Phidias, Lysippus, and others who obtained celebrity in their art. This arose from their being employed by great cities, by kings, or by wealthy citizens. Now others, who, not less studious of their art, nor less endued with great genius and skill, did not enjoy equal fame, because employed by persons of lower rank and of slenderer means, and not from their unskilfulness, seem to have been deserted by fortune; such were Hellas the Athenian, Chion of Corinth, Myagrus the Phoc?an, Pharax the Ephesian, Bedas of Byzantium, and many more; among the Painters, Aristomenes of Thasos, Polycles of Adramyttium, Nicomachus and others, who were wanting neither in industry, study of their art, nor talent. But their poverty, the waywardness of fortune, or their ill success in competition with others, prevented their advancement.

3. Nor can we wonder that from the ignorance of the public in respect of art many skilful artists remain in obscurity; but it is scandalous that friendship and connexion should lead men, for their sake, to give partial and untrue opinions. If, as Socrates would have had it, every one''s feelings, opinions, and information in science could be open to view, neither favour nor ambition would prevail, but those, who by study and great learning acquire the greatest knowledge, would be eagerly sought after. Matters are not however in this state as they ought to be, the ignorant rather than the learned being successful, and as it is never worth while to dispute with an ignorant man, I propose to shew in these precepts the excellence of the science I profess.

4. In the first book, O Emperor, I laid before you an explanation of the art, its requisites, and the learning an architect should possess, and I added the reasons why he should possess them. I also divided it into different branches and defined them: then, because chiefest and most necessary, I have explained the proper method of setting out the walls of a city, and obtaining a healthy site for it, and have exhibited in diagrams, the winds, and quarters whence they blow. I have shewn the best methods of laying out the streets and lanes, and thus completed the first book. In the second book I have analysed the nature and qualities of the materials used in building, and adverted to the purposes to which they are best adapted. In this third book I shall speak of the sacred temples of the immortal gods, and explain them particularly.

Chapter 1
1. The design of Temples depends on symmetry, the rules of which Architects should be most careful to observe. Symmetry arises from proportion, which the Greeks call ????????. Proportion is a due adjustment of the size of the different parts to each other and to the whole; on this proper adjustment symmetry depends. Hence no building can be said to be well designed which wants symmetry and proportion. In truth they are as necessary to the beauty of a building as to that of a well formed human figure,

2. which nature has so fashioned, that in the face, from the chin to the top of the forehead, or to the roots of the hair, is a tenth part of the height of the whole body. From the chin to the crown of the head is an eighth part of the whole height, and from the nape of the neck to the crown of the head the same. From the upper part of the breast to the roots of the hair a sixth; to the crown of the head a fourth. A third part of the height of the face is equal to that from the chin to under side of the nostrils, and thence to the middle of the eyebrows the same; from the last to the roots of the hair, where the forehead ends, the remaining third part. The length of the foot is a sixth part of the height of the body. The fore-arm a fourth part. The width of the breast a fourth part. Similarly have other members their due proportions, by attention to which the ancient Painters and Sculptors obtained so much reputation.

3. Just so the parts of Temples should correspond with each other, and with the whole. The navel is naturally placed in the centre of the human body, and, if in a man lying with his face upward, and his hands and feet extended, from his navel as the centre, a circle be described, it will touch his fingers and toes. It is not alone by a circle, that the human body is thus circumscribed, as may be seen by placing it within a square. For measuring from the feet to the crown of the head, and then across the arms fully extended, we find the latter measure equal to the former; so that lines at right angles to each other, enclosing the figure, will form a square.

4. If Nature, therefore, has made the human body so that the different members of it are measures of the whole, so the ancients have, with great propriety, determined that in all perfect works, each part should be some aliquot part of the whole; and since they direct, that this be observed in all works, it must be most strictly attended to in temples of the gods, wherein the faults as well as the beauties remain to the end of time.

5. It is worthy of remark, that the measures necessarily used in all buildings and other works, are derived from the members of the human body, as the digit, the palm, the foot, the cubit, and that these form a perfect number, called by the Greeks ???????. The ancients considered ten a perfect number, because the fingers are ten in number, and the palm is derived from them, and from the palm is derived the foot. Plato, therefore, called ten a perfect number, Nature having formed the hands with ten fingers, and also because it is composed of units called ??????? in Greek, which also advancing beyond ten, as to eleven, twelve, &c. cannot be perfect until another ten are included, units being the parts whereof such numbers are composed.

6. The mathematicians, on the other hand, contend for the perfection of the number six, because, according to their reasoning, its divisors equal its number: for a sixth part is one, a third two, a half three, two-thirds four, which they call ????????; the fifth in order, which they call ???????????, five, and then the perfect number six. When it advances beyond that, a sixth being added, which is called ???????, we have the number seven. Eight are formed by adding a third, called triens, and by the Greeks, ?????????. Nine are formed by the addition of a half, and thence called sesquilateral; by the Greeks ????????; if we add the two aliquot parts of it, which form ten, it is called bes alterus, or in Greek ???????????. The number eleven, being compounded of the original number, and the fifth in order is called ??????????????. The number twelve, being the sum of the two simple numbers, is called ?????????.

7. Moreover, as the foot is the sixth part of a man''s height, they contend, that this number, namely six, the number of feet in height, is perfect: the cubit, also, being six palms, consequently consists of twenty-four digits. Hence the states of Greece appear to have divided the drachma, like the cubit, that is into six parts, which were small equal sized pieces of brass, similar to the asses, which they called oboli; and, in imitation of the twenty-four digits, they divided the obolus into four parts, which some call dichalca, others trichalca.

8. Our ancestors, however, were better pleased with the number ten, and hence made the denarius to consist of ten brass asses, and the money to this day retains the name of denarius. The sestertius, a fourth part of a denarius, was so called, because composed of two asses, and half of another. Thus finding the numbers six and ten perfect, they added them together, and formed sixteen, a still more perfect number. The foot measure gave rise to this, for subtracting two palms from the cubit, four remains, which is the length of a foot; and as each palm contains four digits, the foot will consequently contain sixteen, so the denarius was made to contain an equal number of asses.

9. If it therefore appear, that numbers had their origin from the human body, and proportion is the result of a due adjustment of the different parts to each other, and to the whole, they are especially to be commended, who, in designing temples to the gods, so arrange the parts that the whole may harmonize in their proportions and symmetry.

Chapter 2
1. The principles of temples are distinguished by their different forms. First, that known by the appellation IN ANTIS, which the Greeks call ???? ?? ?????????; then the PROSTYLOS, PERIPTEROS, PSEUDODIPTEROS, DIPTEROS, HYP?THROS. Their difference is as follows.

2. A temple is called IN ANTIS, when it has ant? or pilasters in front of the walls which enclose the cell, with two columns between the ant?, and crowned with a pediment, proportioned as we shall hereafter direct. There is an example of this species of temple, in that of the three dedicated to Fortune, near the Porta Collina.

3. The PROSTYLOS temple is similar, except that it has columns instead of ant? in front, which are placed opposite to ant? at the angles of the cell, and support the entablature, which returns on each side as in those in antis. An example of the prostylos exists in the temple of Jupiter and Faunus, in the island of the Tyber.

4. The AMPHIPROSTYLOS is similar to the prostylos, but with this addition, that the columns and pediment in the front are repeated in the rear of the temple.

5. The PERIPTEROS has six columns in the front and rear, and eleven on the flanks, counting in the two columns at the angles, and these eleven are so placed that their distance from the wall is equal to an intercolumniation, or space between the columns all round, and thus is formed a walk around the cell of the temple, such as may be seen in the portico of the theatre of Metellus, in that of Jupiter Stator, by Hermodus, and in the temple of Honour and Virtue without a POSTICUM designed by Mutius, near the trophy of Marius.

6. The PSEUDODIPTEROS is constructed with eight columns in front and rear, and with fifteen on the sides, including those at the angles. The walls of the cell are opposite to the four middle columns of the front and of the rear. Hence from the walls to the front of the lower part of the columns, there will be an interval equal to two intercolumniations and the thickness of a column all round. No example of such a temple is to be found in Rome, but of this sort was the temple of Diana, in Magnesia, built by Hermogenes of Alabanda, and that of Apollo, by Menesthes.

7. The DIPTEROS is octastylos like the former, and with a pronaos and posticum, but all round the cell are two ranks of columns. Such are the Doric temple of Quirinus, and the temple of Diana at Ephesus, built by Ctesiphon.

8. The HYP?THROS is decastylos, in the pronaos and posticum. In other respects it is similar to the dipteros, except that in the inside it has two stories of columns all round, at some distance from the walls, after the manner of the peristylia of porticos. The middle of the interior part of the temple is open to the sky, and it is entered by two doors, one in front and the other in the rear. Of this sort there is no example at Rome, there is, however, an octastyle specimen of it at Athens, the temple of Jupiter Olympius.

Chapter 3
1. There are five species of temples, whose names are, PYCNOSTYLOS, that is, thick set with columns: SYSTYLOS, in which the columns are not so close: DIASTYLOS, where they are still wider apart: AR?OSTYLOS, when placed more distant from each other than in fact they ought to be: EUSTYLOS, when the intercolumniation, or space between the columns, is of the best proportion.

2. PYCNOSTYLOS, is that arrangement wherein the columns are only once and a half their thickness apart, as in the temple of the god Julius, in that of Venus in the forum of C?sar, and in other similar buildings. SYSTYLOS, is the distribution of columns with an intercolumniation of two diameters: the distance between their plinths is then equal to their front faces. Examples of it are to be seen in the temple of Fortuna Equestris, near the stone theatre, and in other places.

3. This, no less than the former arrangement, is faulty; because matrons, ascending the steps to supplicate the deity, cannot pass the intercolumniations arm in arm, but are obliged to enter after each other; the doors are also hidden, by the closeness of the columns, and the statues are too much in shadow. The passages moreover round the temple are inconvenient for walking.

4. DIASTYLOS has intercolumniations of three diameters, as in the temple of Apollo and Diana. The inconvenience of this species is, that the epistylia or architraves over the columns frequently fail, from their bearings being too long.

5. In the AR?OSTYLOS the architraves are of wood, and not of stone or marble; the different species of temples of this sort are clumsy, heavy roofed, low and wide, and their pediments are usually ornamented with statues of clay or brass, gilt in the Tuscan fashion. Of this species is the temple of Ceres, near the Circus Maximus, that of Hercules, erected by Pompey, and that of Jupiter Capitolinus.

6. We now proceed to the EUSTYLOS, which is preferable, as well in respect of convenience, as of beauty and strength. Its intercolumniations are of two diameters and a quarter. The center intercolumniation, in front and in the posticum, is three diameters. It has not only a beautiful effect, but is convenient, from the unobstructed passage it affords to the door of the temple, and the great room allowed for walking round the cell.

7. The rule for designing it is as follows. The extent of the front being given, it is, if tetrastylos, to be divided into eleven parts and a half, not including the projections of the base and plinth at each end: if hexastylos, into eighteen parts: if octastylos, into twenty-four parts and a half. One of either of these parts, according to the case, whether tetrastylos, hexastylos, or octastylos, will be a measure equal to the diameter of one of the columns. Each intercolumniation, except the middle one, front and rear, will be equal to two of these measures and one quarter, and the middle intercolumniation three. The heights of the columns will be eight parts and a half. Thus the intercolumniations and the heights of the columns will have proper proportions.

8. There is no example of eustylos in Rome; but there is one at Teos in Asia, which is octastylos, and dedicated to Bacchus. Its proportions were discovered by Hermogenes, who was also the inventor of the octastylos or pseudodipteral formation. It was he who first omitted the inner ranges of columns in the dipteros, which, being in number thirty-eight, afforded the opportunity of avoiding considerable expense. By it a great space was obtained for walking all round the cell, and the effect of the temple was not injured because the omission of the columns was not perceptible; neither was the grandeur of the work destroyed.

9. The pteromata, or wings, and the disposition of columns about a temple, were contrived for the purpose of increasing the effect, by the varied appearance of the returning columns, as seen through the front intercolumniations, and also for providing plenty of room for the numbers frequently detained by rain, so that they might walk about, under shelter, round the cell. I have been thus particular on the pseudodipteros, because it displays the skill and ingenuity with which Hermogenes designed those his works; which cannot be but acknowledged as the sources whence his successors have derived their best principles.

10. In ar?ostyle temples the diameter of the columns must be an eighth part of the height. In diastylos, the height of the columns is to be divided into eight parts and a half; one of which is to be taken for the diameter of the column. In systylos, let the height be divided into nine parts and a half; one of those parts will be the diameter of a column. In pycnostylos, one-tenth part of the height is the diameter of the columns. In the eustylos, as well as in the diastylos, the height of the columns is divided into eight parts and a half; one of which is to be taken for the thickness of the column. These, then, are the rules for the several intercolumniations.

11. For, as the distances between the columns increase, so must the shafts of the columns increase in thickness. If, for instance, in the ar?ostylos, they were a ninth or a tenth part of the height, they would appear too delicate and slender; because the air interposed between the columns destroys and apparently diminishes, their thickness. On the other hand, if, in the pycnostylos, their thickness or diameter were an eighth part of the height, the effect would be heavy and unpleasant, on account of the frequent repetition of the columns, and the smallness of the intercolumniations. The arrangement is therefore indicated by the species adopted. Columns at the angles, on account of the unobstructed play of air round them, should be one-fiftieth part of a diameter thicker than the rest, that they may have a more graceful effect. The deception which the eye undergoes should be allowed for in execution.

12. The diminution of columns taken at the hypotrachelium, is to be so ordered, that for columns of fifteen feet and under, it should be one-sixth of the lower diameter. From fifteen to twenty feet in height, the lower diameter is to be divided into six parts and a half; and five parts and a half are to be assigned for the upper thickness of the column. When columns are from twenty to thirty feet high, the lower diameter of the shaft must be divided into seven parts, six of which are given to the upper diameter. From thirty to forty feet high, the lower diameter is divided into seven parts and a half, and six and a half given to the top. From forty to fifty feet, the lower diameter of the shaft is to be divided into eight parts, seven of which must be given to the thickness under the hypotrachelium. If the proportion for greater heights be required, the thickness at top must be found after the preceding method;

13. always remembering, that as the upper parts of columns are more distant from the eye, they deceive it when viewed from below, and that we must, therefore, actually add what they apparently lose. The eye is constantly seeking after beauty; and if we do not endeavour to gratify it by proper proportions and an increase of size, where necessary, and thus remedy the defect of vision, a work will always be clumsy and disagreeable. Of the swelling which is made in the middle of columns, which the Greeks call ???????, so that it may be pleasing and appropriate, I shall speak at the end of the book.

Chapter 4
1. If solid ground can be come to, the foundations should go down to it and into it, according to the magnitude of the work, and the substruction should be built up as solid as possible. Above the ground the wall should be one-half thicker than the columns it is to receive, so that lower parts which carry the greatest weight, may be stronger than the upper part, which is called the stereobata: nor must the mouldings of the bases of the columns project beyond the solid. Thus, also, should be regulated the thickness of all walls above ground. The intervals between the foundations brought up under the columns, should be either rammed down hard, or arched, so as to prevent the foundation piers from swerving.

2. If solid ground cannot be come to, and the ground be loose or marshy, the place must be excavated, cleared, and either alder, olive, or oak piles, previously charred, must be driven with a machine, as close to each other as possible, and the intervals, between the piles, filled with ashes. The heaviest foundations may be laid on such a base.

3. When they are brought up level, the stylobat? (plinths) are placed thereon, according to the arrangement used, and above described for the pycnostylos, systylos, diastylos or eustylos, as the case may be. In the ar?ostylos it is only necessary to preserve, in a peripteral building, twice the number of intercolumniations on the flanks that there are in front, so that the length may be twice the breadth. Those who use twice the number of columns for the length, appear to err, because they thus make one intercolumniation more than should be used.

4. The number of steps in front should always be odd, since, in that case, the right foot, which begins the ascent, will be that which first alights on the landing of the temple. The thickness of the steps should not, I think, be more than ten inches, nor less than nine, which will give an easy ascent. The treads not less than one foot and a half, nor more than two feet; and if the steps are to go all round the temple, they are to be formed in the same manner.

5. But if there is to be a podium on three sides of the temple, the plinths, bases of the columns, columns, coron?, and cymatium, may accord with the stylobata, under the bases of the columns. The stylobata should be so adjusted, that, by means of small steps or stools, it may be highest in the middle. For if it be set out level, it will have the appearance of having sunk in the centre. The mode of adjusting the steps (scamilli impares), in a proper manner, will be shewn at the end of the book.

Chapter 5
1. The scamilli being prepared and set, the bases of the columns may be laid, their height being equal to the semidiameter of the column including the plinth, and their projection, which the Greeks call ??????, one quarter of the diameter of the column. Thus the height and breadth, added together, will amount to one diameter and a half.

2. If the attic base be used, it must be so subdivided that the upper part be one-third of the thickness of the column, and that the remainder be assigned for the height of the plinth. Excluding the plinth, divide the height into four parts, one of which is to be given to the upper torus; then divide the remaining three parts into two equal parts, one will be the height of the lower torus, and the other the height of the scotia, with its fillets, which the Greeks call ???????? (trochilus).

3. If Ionic, they are to be set out so that the base may each way be equal to the thickness and three eighths of the column. Its height and that of the plinth the same as the attic base. The plinth is the same height as in that of the attic base, the remainder, which was equal to one-third part of the column''s diameter, must be divided into seven parts, three of which are given to the upper torus; the remaining four parts are to be equally divided into two, one of which is given to the upper cavetto, with its astragals and listel, the other to the lower cavetto, which will have the appearance of being larger, from its being next to the plinth. The astragals must be an eight part of the scotia, and the whole base on each side is to project three sixteenths of a diameter.

4. The bases being thus completed, we are to raise the columns on them. Those of the pronaos and posticum are to be set up with their axes perpendicular, the angular ones excepted, which, as well as those on the flanks, right and left, are to be so placed that their interior faces towards the cell be perpendicular. The exterior faces will diminish upwards, as above-mentioned. Thus the diminution will give a pleasing effect to the temple.

5. The shafts of the columns being fixed, the proportions of the capitals are thus adjusted: if pillowed, as in the Ionic, they must be so formed that the length and breadth of the abacus be equal to the diameter of the lower part of the column and one eighteenth more, and the height of the whole, including the volutes, half a diameter. The face of the volutes is to recede within the extreme projection of the abacus one thirty-ninth part of the width of the abacus. Having set out these points on the listel of the abacus at the four angles, let fall vertical lines. These are called catheti. The whole height of the capital is now to be divided into nine parts and a half, whereof one part and a half is the height of the abacus, and the remaining eight are for the eye of the volute.

6. Within the line dropt from the angle of the abacus, at the distance of one and a half of the parts last found, let fall another vertical line, and so divide it that four parts and a half being left under the abacus, the point which divides them from the remaining three and a half, may be the centre of the eye of the volute; from which, with a radius equal to one half of one of the parts, if a circle be described, it will be the sixth of the eye of the volute. Through its centre let an horizontal line be drawn, and beginning from the upper part of the vertical diameter of the eye as a centre, let a quadrant be described whose upper part shall touch the under side of the abacus; ten changing the centre, with a radius less than the last by half the width of the diameter of the eye, proceed with other quadrants, so that the last will fall into the eye itself, which happen in the vertical line, at a point perpendicularly under that of setting out.

7. The heights of the parts of the capital are to be so regulated that three of the nine parts and a half, into which it was divided, lie below the level of the astragal on the top of the shaft. The remaining parts are for the cymatium, abacus, and channel. The projection of the cymatium beyond the abacus is not to be greater than the size of the diameter of the eye. The bands of the pillows project beyond the abacus, according to the following rule. Place one point of the compasses in the centre of the eye, and let the other extend to the top of the cymatium, then describing a semicircle, its extreme part will equal the projection of the band of the pillow. The centres, from which the volute is described, should not be more distant from each other than the thickness of the eye, nor the channels sunk more than a twelfth part of their width. The foregoing are the proportions for the capitals of columns which do not exceed fifteen feet in height: when they exceed that, they must be otherwise proportioned, though upon similar principles, always observing that the square of the abacus is to be a ninth part more than the diameter of the column, so that, inasmuch as its diminution is less as its height is greater, the capital which crowns it may also be augmented in height and projection.

8. The method of describing volutes, in order that they may be properly turned and proportioned, will be given at the end of the book. The capitals being completed, and set on tops of the shafts, not level throughout the range of columns, but so arranged with a gauge as to follow the inclination which the small steps on the stylobata produce, which must be added to them on the central part of the top of the abacus, that the regularity of the epistylia may be preserved: we may now consider the proportion of these epistylia, or architraves. When the columns are at least twelve and not more than fifteen feet high, the architrave must be half a diameter in height. When they are from fifteen to twenty feet in height, the height of the column is to be divided into thirteen parts, and one of them taken for the height of the architrave. So from twenty to twenty-five feet, let the height be divided into twelve parts and a half, and one part be taken for the height of the architrave. Thus, in proportion to the height of the column, is the architrave to be proportioned;

9. always remembering, that the higher the eye has to reach, the greater is the difficulty it has in piercing the density of the air, its power being diminished as the height increases; of which the result is, a confusion of the image. Hence, to preserve a sensible proportion of parts, if in high situations, or of colossal dimensions, we must modify them accordingly, so that they may appear of the size intended. The under side of the architrave is to be as wide as the upper diameter of the column, at the part under the capital; its upper part equal in width to the lower diameter of the column.

10. Its cymatium is to be one seventh part of the whole height, and its projection the same. After the cymatium is taken out, the remainder is to be divided into twelve parts, three of which are to be given to the lower fascia, four to the next, and five to the upper one. The zophorus, or frieze, is placed over the epistylium, than which it must be one fourth less in height; but if sculptured, it must be one fourth part higher, that the effect of the carving may not be injured. Its cymatium is to be a seventh part of its height, the projection equal to the height.

11. Above the frieze is placed the dentil-band, whose height must be equal to that of the middle fascia of the architrave, its projection equal to its height. The cutting thereof, which the Greeks call ?????? (metoche), is to be so executed that the width of each dentil may be half its height, and the space between them two-thirds of the width of a dentil. The cymatium is to be one sixth part of its height. The corona, with its cymatium, but without the sima is to be the same height as the middle fascia of the architrave. The projection of the corona and dentils, together is to be equal to the height from the frieze to the top of the cymatium of the corona. It may, indeed, be generally observed, that projections are more beautiful when they are equal to the height of the member.

12. The height of the tympanum, which crowns the whole work, is to be equal to one ninth part of the extent of the corona, measured from one extremity of its cymatium to the other, and set up in the centre. Its face is to stand perpendicularly over the architrave and the hypotrachelia of the columns. The coron? over the tympanum are to be equal to that below, without the sim?. Above the coron? are set the sim?, which the Greeks call ??????????, whose height must be one-eighth more than that of the corona. The height of the acroteria is to be equal to that of the middle of the tympanum; the central ones one eighth part higher than those at the angles.

13. All members over the capitals of columns, such as architraves, friezes, coron?, tympana, crowning members (fastigia), and acroteria, should not be vertical, but inclined forwards, each a twelfth part of its height; and for this reason, that when two lines are produced from the eye, one to the upper part of a member, and the other to its lower part, the upper line or visual ray will be longer than the lower one, and if really vertical, the member will appear to lean backwards; but if the members are set out as above directed, they will have the appearance of being perpendicular.

14. The number of flutes in a column is twenty-four. They are to be hollowed, so that a square kept passing round their surface, and at the same time kept close against the arrises of the fillets, will touch some point in their circumference and the arrises themselves throughout its motion. The additional thickness of the flutes and fillets in the middle of the column, arising from the entasis or swelling, will be proportional to the swelling.

15. On the sim? of the coron? on the sides of temples, lions'' heads should be carved; and they are to be so disposed that one may come over each column, and the others at equal distances from each other, and answering to the middle of each tile. Those which are placed over the columns are to be bored through, so as to carry off the rain-water collected in the gutter. But the intermediate ones must be solid, so that the water from the tiles, which is collected in the gutter, may not be carried off in the intercolumniations, and fall on those passing. Those over the columns will appear to vomit forth streams of water from their mounts. In this book I have done my utmost to describe the proportions of Ionic temples: in that following I shall explain the proportions of Doric and Corinthian temples.

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Thayer''s Notes:
a these form a perfect number: The basic idea — that man is the module of architecture — is all one need retain of this hash. For those, however, who may delight in explanations and the byways they afford, here is mine.

In the Roman system of measurement, there were 4 digits to a palm, 4 palms to a foot, and 1 1/2 feet to a cubit. All four of these units are based on the human body: a digit (or finger) is indeed roughly the thickness of a finger; a palm is the width of the palm, which is of course four fingers; a foot is a foot; and a cubit is a forearm, from the elbow to the tip of the fingers. These measures were well nigh universal in ancient societies. (We''re ignoring here the unrelated inch, 12 to a foot, apparently little used in architecture and engineering.)

Left at that, we would have a simple explanation of architectural metrology, that may be considered noble and "perfect": any structure built based on these measurements can be viewed as a projection of the human body. (Lest we find this primitive somehow, believing that we moderns, in adopting the metric system, have dispensed with such crudities: a reminder that the basic module of the metric system as originally defined, the meter, was one ten-millionth of a meridian of the earth from the pole to the equator. We have merely shifted from the human foot to the earth it treads, from biology to geography, and measurement remains a metaphor.)

Now architecture as Vitruvius knew it was essentially a creation of the Greeks, and they discovered yet another idea, equally noble and marvelous: that the universe is mathematics. (That this idea too has remained with us, indeed intensified, it is less needful to point out.)

What Vitruvius has done in the passage above is to graft this second notion onto the first one. He''s done it unfortunately awkwardly. Step by step:

The ancients considered ten a perfect number, because the fingers are ten in number: pretty much universal in ancient societies, and in some sense true.

and the palm is derived from them: yes and no, to the extent that the width of four fingers of one hand is derived from the fact that we have two sets like this plus a couple of thumbs.

and from the palm is derived the foot: palpable nonsense, if you''ll pardon the pun.

[ten] is composed of units called monads in Greek, which also advancing beyond ten, as to eleven, twelve, &c. cannot be perfect until another ten are included: this is some very ill digested Pythagorean arithmetic, yielding a platitude that begs the question. All it means is that ten is perfect, because (ten being perfect in the first place) you have to add ten ones to it to get the next number divisible by ten. . .!
The followers of Pythagoras would have been horrified, of course: they saw the perfection of ten as springing from a geometrical figure they called the tetractys (1 + 2 + 3 + 4 = 10), one of the so?called triangular numbers, which do indeed have very interesting, almost magical properties, closely related to the series that fascinated Fibonacci, and the triangles discovered by Pascal, and to many profound developments in number theory to our own day — but of all this, our writer seems quite unaware.

The mathematicians, on the other hand, contend for the perfection of the number six, because, according to their reasoning, its divisors equal its number: here Vitruvius consciously shifts gears — "on the other hand", as he says — to a different part of number theory.

The Greeks had discovered that a very few numbers had the peculiar property that the sum of all their prime divisors was equal to the number itself: the first such number is 6, because 6 can be divided by 1, 2 and 3, and these add up to 6 again. They called such numbers "perfect", because they contain themselves, as it were: other numbers being deficient (10 for example, whose factors 1, 2 and 5 only add to 8) or excessive (12 for example, whose factors 1, 2, 3, 4 and 6 add to 16).

Well that''s very nice, but what does that have to do with 10? Nothing, of course. 10 is not a perfect number in this system. (But hold that 10 in your head for a moment anyway.)

for a sixth part is one, a third two, a half three: back to the perfection of the number 6, this might be a rather elegantly chiastic exposition of 1 + 2 + 3 = 6 —

except that Vitruvius proceeds to spoil it altogether by adding two-thirds four . . .: and the rest of that long paragraph is utterly and completely irrelevant to the perfection of numbers, although it does apply to the terminology of ancient music and prosody. (About two weeks after I wrote this, while entering the Latin text, I was delighted to see that the Teubner edition brackets this passage starting at the exact same point, footnoting that in the editor''s opinion, this is not Vitruvius writing at all, rather the addition of some medieval school pedant.)

Skipping a bit, in which we are told things about the number 24 and the number 10 (fine), we come to the clincher, showing that our writer was no theoretical mathematician: Thus finding the numbers six and ten perfect, they added them together, and formed sixteen, a still more perfect number. Hell-bent on finding 16 (remember the digits in the foot?) somewhere, he adds 6 and 10. . . and declares the result "more perfect" than either, when it is of course not perfect in any system, i.e., it is neither triangular nor the sum of its divisors, nor is it naturally found in the human body. At this point, appearing satisfied, although in fact I have a feeling that he''s more than dimly aware of the sleight-of?foot, our author drops it all very quickly: so much the better.

Vitruvius has demonstrated to us that his great gifts lay on the practical, not the theoretical, side of architecture.

Not to leave any loose ends, for the best graft ever attempted of the two great ideas (here so imperfectly glued together), an extraordinary poetic synthesis of biology and mathematics, springing like an orderly forest of trees from an observation of Quintilian''s, see The Garden of Cyrus and its shadow side Hydriotaphia or Urn Burial, by Sir Thomas Browne (1605?1682).

b that of the three dedicated to Fortune: I find this example and the entire catalogue of examples adduced by Vitruvius sadly moving. Vitruvius'' purpose in giving these examples was of course to illustrate a technical concept by pointing to buildings everyone knew.

Not one of them still exists today.

Vitruvius'' use of them as examples is turned on its head by us: scholars use this text to attempt to get the barest shred of information about them. (For the three temples of Fortune, see this article in Platner.)

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