Ten Books on Architecture/Book III
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
69 |
On Symmetry: in Temples and in the Human Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
72 |
Classification of Temples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
75 |
The Proportions of Intercolumniations and of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
78 |
The Foundations and Substructures of Temples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
86 |
Proportions of The Base, Capitals, and Entablature in the Ionic Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
90 |
BOOK III
INTRODUCTION
1. Apollo at Delphi, through the oracular utterance of his priestess, pronounced Socrates the wisest of men. Of him it is related that he said with sagacity and great learning that the human breast should have been furnished with open windows, so that men might not keep their feelings concealed, but have them open to the view. Oh that nature, following his idea, had constructed them thus unfolded and obvious to the view! For if it had been so, not merely the virtues and vices of the mind would be easily visible, but also its knowledge of branches of study, displayed to the contemplation of the eyes, would not need testing by untrustworthy powers of judgement, but a singular and lasting influence would thus be lent to the learned and wise. However, since they are not so constructed, but are as nature willed them to be, it is impossible for men, while natural abilities are concealed in the breast, to form a judgement on the quality of the knowledge of the arts which is thus deeply hidden. And if artists themselves testify to their own skill, they can never, unless they are wealthy or famous from the age of their studios, or unless they are also possessed of the public favour and of eloquence, have an influence commensurate with their devotion to their pursuits, so that people may believe them to have the knowledge which they profess to have.
2. In particular we can learn this from the case of the sculptors and painters of antiquity. Those among them who were marked by high station or favourably recommended have come down to posterity with a name that will last forever; for instance, Myron, Polycletus, Phidias, Lysippus, and the others who have attained to fame by their art. For they acquired it by the execution of works for great states or for kings or for citizens of rank. But those who, being men of no less enthusiasm, natural ability, and dexterity than those famous artists, and who executed no less perfectly finished works for citizens of low station, are unremembered, not because they lacked diligence or dexterity in their art, but because fortune failed them; for instance, Teleas of Athens, Chion of Corinth, Myager the Phocaean, Pharax of Ephesus, Boedas of Byzantium, and many others. Then there were painters like Aristomenes of Thasos, Polycles and Andron of Ephesus, Theo of Magnesia, and others who were not deficient in diligence or enthusiasm for their art or in dexterity, but whose narrow means or ill-luck, or the higher position of their rivals in the struggle for honour, stood in the way of their attaining distinction.
3. Of course, we need not be surprised if artistic excellence goes unrecognized on account of being unknown; but there should be the greatest indignation when, as often, good judges are flattered by the charm of social entertainments into an approbation which is a mere pretence. Now if, as Socrates wished, our feelings, opinions, and knowledge gained by study had been manifest and clear to see, popularity and adulation would have no influence, but men who had reached the height of knowledge by means of correct and definite courses of study, would be given commissions without any effort on their part. However, since such things are not plain and apparent to the view, as we think they should have been, and since I observe that the uneducated rather than the educated are in higher favour, thinking it beneath me to engage with the uneducated in the struggle for honour, I prefer to show the excellence of our department of knowledge by the publication of this treatise.
4. In my first book, Emperor, I described to you the art, with its points of excellence, the different kinds of training with which the architect ought to be equipped, adding the reasons why he ought to be skilful in them, and I divided up the subject of architecture as a whole among its departments, duly defining the limits of each. Next, as was preeminent and necessary, I explained on scientific principles the method of selecting healthy sites for fortified towns, pointed out by geometrical figures the different winds and the quarters from which they blow, and showed the proper way to lay out the lines of streets and rows of houses within the walls. Here I fixed the end of my first book. In the second, on building materials, I treated their various advantages in structures, and the natural properties of which they are composed. In this third book I shall speak of the temples of the immortal gods, describing and explaining them in the proper manner.
CHAPTER I
ON SYMMETRY: IN TEMPLES AND IN THE HUMAN BODY
1. The design of a temple depends on symmetry, the principles of which must be most carefully observed by the architect. They are due to proportion, in Greek άναλογία. Proportion is a correspondence among the measures of the members of an entire work, and of the whole to a certain part selected as standard. From this result the principles of symmetry. Without symmetry and proportion there can be no principles in the design of any temple; that is, if there is no precise relation between its members, as in the case of those of a well shaped man.
2. For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. If we take the height of the face itself, the distance from the bottom of the chin to the under side of the nostrils is one third of it; the nose from the under side of the nostrils to a line between the eyebrows is the same; from there to the lowest roots of the hair is also a third, comprising the forehead. The length of the foot is one sixth of the height of the body; of the forearm, one fourth; and the breadth of the breast is also one fourth. The other members, too, have their own symmetrical proportions, and it was by employing them that the famous painters and sculptors of antiquity attained to great and endless renown.
3. Similarly, in the members of a temple there ought to be the greatest harmony in the symmetrical relations of the different parts to the general magnitude of the whole. Then again, in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centred at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square.
4. Therefore, since nature has designed the human body so that its members are duly proportioned to the frame as a whole, it appears that the ancients had good reason for their rule, that in perfect buildings the different members must be in exact symmetrical relations to the whole general scheme. Hence, while transmitting to us the proper arrangements for buildings of all kinds, they were particularly careful to do so in the case of temples of the gods, buildings in which merits and faults usually last forever.
5. Further, it was from the members of the body that they derived the fundamental ideas of the measures which are obviously necessary in all works, as the finger, palm, foot, and cubit. These they apportioned so as to form the "perfect number," called in Greek τέλειον, and as the perfect number the ancients fixed upon ten. For it is from the number of the fingers of the hand that the palm is found, and the foot from the palm. Again, while ten is naturally perfect, as being made up by the fingers of the two palms, Plato also held that this number was perfect because ten is composed of the individual units, called by the Greeks μονάδες. But as soon as eleven or twelve is reached, the numbers, being excessive, cannot be perfect until they come to ten for the second time; for the component parts of that number are the individual units.
6. The mathematicians, however, maintaining a different view, have said that the perfect number is six, because this number is composed of integral parts which are suited numerically to their method of reckoning: thus, one is one sixth; two is one third; three is one half; four is two thirds, or δίμοιρος as they call it; five is five sixths, called πεντάμοιρος; and six is the perfect number. As the number goes on growing larger, the addition of a unit above six is the εφεκτος; eight, formed by the addition of a third part of six, is the integer and a third, called επιτριτος; the addition of one half makes nine, the integer and a half, termed ημιόλιος; the addition of two thirds, making the number ten, is the integer and two thirds, which they call επιδιμοιρος; in the number eleven, where five are added, we have the five sixths, called επιπεμπτος; finally, twelve, being composed of the two simple integers, is called διπλάσιος.
7. And further, as the foot is one sixth of a man's height, the height of the body as expressed in number of feet being limited to six, they held that this was the perfect number, and observed that the cubit consisted of six palms or of twenty-four fingers. This principle seems to have been followed by the states of Greece. As the cubit consisted of six palms, they made the drachma, which they used as their unit, consist in the same way of six bronze coins, like our asses, which they call obols; and, to correspond to the fingers, divided the drachma into twenty-four quarter-obols, which some call dichalca others trichalca.
8. But our countrymen at first fixed upon the ancient number and made ten bronze pieces go to the denarius, and this is the origin of the name which is applied to the denarius to this day. And the fourth part of it, consisting of two asses and half of a third, they called "sesterce." But later, observing that six and ten were both of them perfect numbers, they combined the two, and thus made the most perfect number, sixteen. They found their authority for this in the foot. For if we take two palms from the cubit, there remains the foot of four palms; but the palm contains four fingers. Hence the foot contains sixteen fingers, and the denarius the same number of bronze asses.
9. Therefore, if it is agreed that number was found out from the human fingers, and that there is a symmetrical correspondence between the members separately and the entire form of the body, in accordance with a certain part selected as standard, we can have nothing but respect for those who, in constructing temples of the immortal gods, have so arranged the members of the works that both the separate parts and the whole design may harmonize in their proportions and symmetry.
CHAPTER II
CLASSIFICATION OF TEMPLES
1. There are certain elementary forms on which the general aspect of a temple depends. First there is the temple in antis, or ναὸς ἐν παραστάσιν as it is called in Greek; then the prostyle, amphiprostyle, peripteral, pseudodipteral, dipteral, and hypaethral. These different forms may be described as follows.
2. It will be a temple in antis when it has antae carried out in front of the walls which enclose the cella, and in the middle, between the antae, two columns, and over them the pediment constructed in the symmetrical proportions to be described later in this work. An example will be found at the Three Fortunes, in that one of the three which is nearest the Colline gate.
3. The prostyle is in all respects like the temple in antis, except that at the corners, opposite the antae, it has two columns, and that it has architraves not only in front, as in the case of the temple in antis, but also one to the right and one to the left in the wings. An example of this is the temple of Jove and Faunus in the Island of the Tiber.
4. The amphiprostyle is in all other respects like the prostyle, but has besides, in the rear, the same arrangement of columns and pediment.
5. A temple will be peripteral that has six columns in front and six in the rear, with eleven on each side including the corner 
THE CLASSIFICATION OF TEMPLES ACCORDING TO THE ARRANGEMENTS OF THE COLONNADES
HYPAETHRAL TEMPLE OF VITRUVIUS COMPARED WITH PARTHENON AND THE TEMPLE OF APOLLO NEAR MILETUS
6. The pseudodipteral is so constructed that in front and in the rear there are in each case eight columns, with fifteen on each side, including the corner columns. The walls of the cella in front and in the rear should be directly over against the four middle columns. Thus there will be a space, the width of two intercolumniations plus the thickness of the lower diameter of a column, all round between the walls and the rows of columns on the outside. There is no example of this in Rome, but at Magnesia there is the temple of Diana by Hermogenes, and that of Apollo at Alabanda by Mnesthes.
7. The dipteral also is octastyle in both front and rear porticoes, but it has two rows of columns all round the temple, like the temple of Quirinus, which is Doric, and the temple of Diana at Ephesus, planned by Chersiphron, which is Ionic.
8. The hypaethral is decastyle in both front and rear porticoes. In everything else it is the same as the dipteral, but inside it has two tiers of columns set out from the wall all round, like the colonnade of a peristyle. The central part is open to the sky, without a roof. Folding doors lead to it at each end, in the porticoes in front and in the rear. There is no example of this sort in Rome, but in Athens there is the octastyle in the precinct of the Olympian.
CHAPTER III
THE PROPORTIONS OF INTERCOLUMNIATIONS AND OF COLUMNS
1. There are five classes of temples, designated as follows: pycnostyle, with the columns close together; systyle, with the intercolumniations a little wider; diastyle, more open still; araeostyle, farther apart than they ought to be; eustyle, with the intervals apportioned just right. 2. The pycnostyle is a temple in an intercolumniation of which the thickness of a column and a half can be inserted: for example, the temple of the Divine Caesar, that of Venus in Caesar's forum, and others constructed like them. The systyle is a temple in which
CLASSIFICATION OF TEMPLES ACCORDING TO INTERCOLUMNIATION
3. These two kinds have practical disadvantages. When the matrons mount the steps for public prayer or thanksgiving, they cannot pass through the intercolumniations with their arms about one another, but must form single file; then again, the effect of the folding doors is thrust out of sight by the crowding of the columns, and likewise the statues are thrown into shadow; the narrow space interferes also with walks round the temple.
4. The construction will be diastyle when we can insert the thickness of three columns in an intercolumniation, as in the case of the temple of Apollo and Diana. This arrangement involves the danger that the architraves may break on account of the great width of the intervals.
5. In araeostyles we cannot employ stone or marble for the architraves, but must have a series of wooden beams laid upon the columns. And moreover, in appearance these temples are clumsy-roofed, low, broad, and their pediments are adorned in the Tuscan fashion with statues of terra-cotta or gilt bronze: for example, near the Circus Maximus, the temple of Ceres and Pompey's temple of Hercules; also the temple on the Capitol.
6. An account must now be given of the eustyle, which is the most approved class, and is arranged on principles developed with a view to convenience, beauty, and strength. The intervals should be made as wide as the thickness of two columns and a quarter, but the middle intercolumniations, one in front and the other in the rear, should be of the thickness of three columns. Thus built, the effect of the design will be beautiful, there will be no obstruction at the entrance, and the walk round the cella will be dignified.
7. The rule of this arrangement may be set forth as follows. If a tetrastyle is to be built, let the width of the front which shall 
THE EUSTYLE TEMPLE OF VITRUVIUS COMPARED WITH THE TEMPLE OF TEOS
8. We have no example of this in Rome, but at Teos in Asia Minor there is one which is hexastyle, dedicated to Father Bacchus.
These rules for symmetry were established by Hermogenes, who was also the first to devise the principle of the pseudodipteral octastyle. He did so by dispensing with the inner rows of thirtyeight columns which belonged to the symmetry of the dipteral temple, and in this way he made a saving in expense and labour. He thus provided a much wider space for the walk round the cella between it and the columns, and without detracting at all from the general effect, or making one feel the loss of what had been really superfluous, he preserved the dignity of the whole work by his new treatment of it.
9. For the idea of the pteroma and the arrangement of the columns round a temple were devised in order that the intercolumniations might give the imposing effect of high relief; and also, in case a multitude of people should be caught in a heavy shower and detained, that they might have in the temple and round the cella a wide free space in which to wait. These ideas are developed, as I have described, in the pseudodipteral arrangement of a temple. It appears, therefore, that Hermogenes 
VITRUVIUS' RULES FOR THE DIAMETER AND HEIGHT OF COLUMNS IN THE DIFFERENT CLASSES OF TEMPLE COMPARED WITH ACTUAL EXAMPLES
10. In araeostyle temples, the columns should be constructed so that their thickness is one eighth part of their height. In the diastyle, the height of a column should be measured off into eight and a half parts, and the thickness of the column fixed at one of these parts. In the systyle, let the height be divided into nine and a half parts, and one of these given to the thickness of the column. In the pycnostyle, the height should be divided into ten parts, and one of these used for the thickness of the column. In the eustyle temple, let the height of a column be divided, as in the systyle, into nine and a half parts, and let one part be taken for the thickness at the bottom of the shaft. With these dimensions we shall be taking into account the proportions of the intercolumniations.
11. For the thickness of the shafts must be enlarged in proportion to the increase of the distance between the columns. In the araeostyle, for instance, if only a ninth or tenth part is given to the thickness, the column will look thin and mean, because the width of the intercolumniations is such that the air seems to eat away and diminish the thickness of such shafts. On the other hand, in pycnostyles, if an eighth part is given to the thickness, it will make the shaft look swollen and ungraceful, because the intercolumniations are so close to each other and so narrow. We must therefore follow the rules of symmetry required by each kind of building. Then, too, the columns at the corners should be made thicker than the others by a fiftieth of their own diameter, because they are sharply outlined by the unobstructed air round them, and seem to the beholder more slender than they are. Hence, we must counteract the ocular deception by an adjustment of proportions.
12. Moreover, the diminution in the top of a column at the necking seems to be regulated on the following principles: if a column is fifteen feet or under, let the thickness at the bottom 
THE DIMINUTION OF COLUMNS IN RELATION TO THEIR DIMENSIONS OF HEIGHT
13. These proportionate enlargements are made in the thickness of columns on account of the different heights to which the eye has to climb. For the eye is always in search of beauty, and if we do not gratify its desire for pleasure by a proportionate enlargement in these measures, and thus make compensation for ocular deception, a clumsy and awkward appearance will be presented to the beholder. With regard to the enlargement made at the middle of columns, which among the Greeks is called eντασις, at the end of the book a figure and calculation will be subjoined, showing how an agreeable and appropriate effect may be produced by it.
CHAPTER IV
THE FOUNDATIONS AND SUBSTRUCTURES OF TEMPLES
1. The foundations of these works should be dug out of the solid ground, if it can be found, and carried down into solid ground as far as the magnitude of the work shall seem to require, and the whole substructure should be as solid as it can possibly be laid. Above ground, let walls be laid under the columns, thicker by one half than the columns are to be, so that the lower may be
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| 1 | 2 |
| THE ENTASIS OF COLUMNS | |
| 1. The entasis as given by Fra Giocondo in the edition of 1511.
2. The entasis from the temple of Mars Ultor in Rome compared with Vignola's rule for entasis. | |
2. If, however, solid ground cannot be found, but the place proves to be nothing but a heap of loose earth to the very bottom, or a marsh, then it must be dug up and cleared out and set with piles made of charred alder or olive wood or oak, and these must be driven down by machinery, very closely together like bridge-piles, and the intervals between them filled in with charcoal, and finally the foundations are to be laid on them in the most solid form of construction. The foundations having been brought up to the level, the stylobates are next to be put in place.
3. The columns are then to be distributed over the stylobates in the manner above described: close together in the pycnostyle; in the systyle, diastyle, or eustyle, as they are described and arranged above. In araeostyle temples one is free to arrange them as far apart as one likes. Still, in peripterals, the columns should be so placed that there are twice as many intercolumniations on the sides as there are in front; for thus the length of the work will be twice its breadth. Those who make the number of columns double, seem to be in error, because then the length seems to be one intercolumniation longer than it ought to be.
4. The steps in front must be arranged so that there shall always be an odd number of them; for thus the right foot, with which one mounts the first step, will also be the first to reach the level of the temple itself. The rise of such steps should, I think, be limited to not more than ten nor less than nine inches; for then the ascent will not be difficult. The treads of the steps ought to be made not less than a foot and a half, and not more than two feet deep. If there are to be steps running all round the temple, they should be built of the same size.
5. But if a podium is to be built on three sides round the temple, it should be so constructed that its plinths, bases, dies, coronae, and cymatiumare appropriate to the actual stylobate which is to be under the bases of the columns.
FRA GIOCONDO'S IDEA OF THE "SCAMILLI IMPARES"
(From his edition of Vitruvius, Venice, 1511)
The level of the stylobate must be increased along the middle by the scamilli impares; for if it is laid perfectly level, it will look to the eye as though it were hollowed a little. At the end of the book a figure will be found, with a description showing how the scamilli may be made to suit this purpose.
CHAPTER V
PROPORTIONS OP THE BASE, CAPITALS, AND ENTABLATURE IN THE IONIC ORDER
1. This finished, let the bases of the columns be set in place, and constructed in such proportions that their height, including the plinth, may be half the thickness of a column, and their projection (called in Greek ἐκφορά) the same.[1] Thus in both length and breadth it will be one and one half thicknesses of a column.
2. If the base is to be in the Attic style, let its height be so divided that the upper part shall be one third part of the thickness of the column, and the rest left for the plinth. Then, excluding the plinth, let the rest be divided into four parts, and of these let one fourth constitute the upper torus, and let the other three be divided equally, one part composing the lower torus, and the other, with its fillets, the scotia, which the Greeks call τροχίλος.
3. But if Ionic bases are to be built, their proportions shall be so determined that the base may be each way equal in breadth to the thickness of a column plus three eighths of the thickness; its height that of the Attic base, and so too its plinth; excluding the plinth, let the rest, which will be a third part of the thickness of a column, be divided into seven parts. Three of these parts constitute the torus at the top, and the other four are to be divided equally, one part constituting the upper trochilus with its astragals and overhang, the other left for the lower trochilus. But the lower will seem to be larger, because it will project to the edge of the plinth. The astragals must be one eighth of the trochilus. The projection of the base will be three sixteenths of the thickness of a column.
4. The bases being thus finished and put in place, the columns are to be put in place: the middle columns of the front and rear porticoes perpendicular to their own centre; the corner columns, and those which are to extend in a line from them along the sides 
THE IONIC ORDER ACCORDING TO VITRUVIUS COMPARED WITH THE ORDER OF THE MAUSOLEUM AT HALICARNASSUS
The difference between the Roman and the Greek relation of the baluster-side of the capital to the echinus is to be noted.
of the temple to the right and left, are to be set so that their inner sides, which face toward the cella wall, are perpendicular, but their outer sides in the manner which I have described in speaking of their diminution. Thus, in the design of the temple the lines will be adjusted with due regard to the diminution.
5. The shafts of the columns having been erected, the rule for the capitals will be as follows. If they are to be cushion-shaped, they should be so proportioned that the abacus is in length and breadth equivalent to the thickness of the shaft at its bottom plus one eighteenth thereof, and the height of the capital, including the volutes, one half of that amount. The faces of the volutes must recede from the edge of the abacus inwards by one and a half eighteenths of that same amount. Then, the height of the capital is to be divided into nine and a half parts, and down along the abacus on the four sides of the volutes, down along the fillet at the edge of the abacus, lines called "catheti" are to be let fall. Then, of the nine and a half parts let one and a half be reserved for the height of the abacus, and let the other eight be used for the volutes.
6. Then let another line be drawn, beginning at a point situated at a distance of one and a half parts toward the inside from the line previously let fall down along the edge of the abacus. Next, let these lines be divided in such a way as to leave four and a half parts under the abacus; then, at the point which forms the division between the four and a half parts and the remaining three and a half, fix the centre of the eye, and from that centre describe a circle with a diameter equal to one of the eight parts. This will be the size of the eye, and in it draw a diameter on the line of the "cathetus." Then, in describing the quadrants, let the size of each be successively less, by half the diameter of the eye, than that which begins under the abacus, and proceed from the eye until that same quadrant under the abacus is reached.
7. The height of the capital is to be such that, of the nine and a half parts, three parts are below the level of the astragal at the top of the shaft, and the rest, omitting the abacus and the channel, belongs to its echinus. The projection of the echinus beyond the fillet of the abacus should be equal to the size of the eye. The projection of the bands of the cushions should be thus obtained: place one leg of a pair of compasses in the centre of the capital and open out the other to the edge of the echinus; bring this leg round and it will touch the outer edge of the bands. The axes of the volutes should not be thicker than the size of the eye, and the volutes themselves should be channelled out to a depth which is one twelfth of their height. These will be the symmetrical proportions for capitals of columns twenty-five feet high and less. For higher columns the other proportions will be the same, but the length and breadth of the abacus will be the thickness of the lower diameter of a column plus one ninth part thereof; thus, just as the higher the column the less the diminution, so the projection of its capital is proportionately increased and its breadth[2] is correspondingly enlarged.
8. With regard to the method of describing volutes, at the end of the book a figure will be subjoined and a calculation showing how they may be described so that their spirals may be true to the compass.
The capitals having been finished and set up in due proportion to the columns (not exactly level on the columns, however, but with the same measured adjustment, so that in the upper members there may be an increase corresponding to that which was made in the stylobates), the rule for the architraves is to be as follows. If the columns are at least twelve feet and not more than fifteen feet high, let the height of the architrave be equal to half the thickness of a column at the bottom. If they are from fifteen feet to twenty, let the height of a column be measured off into thirteen parts, and let one of these be the height of the architrave. If they are from twenty to twenty-five feet, let this height be divided into twelve and one half parts, and let one of them form the height of the architrave. If they are from twenty-five feet to thirty, let it be divided into twelve parts, and let one of them form the height. If they are higher, the heights of the architraves are to be worked out proportionately in the same manner from the height of the columns.
9. For the higher that the eye has to climb, the less easily can it make its way through the thicker and thicker mass of air. So it fails when the height is great, its strength is sucked out of it, and it conveys to the mind only a confused estimate of the dimensions. Hence there must always be a corresponding increase in the symmetrical proportions of the members, so that whether the buildings are on unusually lofty sites or are themselves somewhat colossal, the size of the parts may seem in due proportion. The depth of the architrave on its under side just above the capital, is to be equivalent to the thickness of the top of the column just under the capital, and on its uppermost side equivalent to the foot of the shaft.
10. The cymatium of the architrave should be one seventh of the height of the whole architrave, and its projection the same. Omitting the cymatium, the rest of the architrave is to be divided into twelve parts, and three of these will form the lowest fascia, four, the next, and five, the highest fascia. The frieze, above the architrave, is one fourth less high than the architrave, but if there are to be reliefs upon it, it is one fourth higher than the architrave, so that the sculptures may be more imposing. Its cymatium is one seventh of the whole height of the frieze, and the projection of the cymatium is the same as its height.
11. Over the frieze comes the line of dentils, made of the same height as the middle fascia of the architrave and with a projection equal to their height. The intersection (or in Greek μετόπη) is apportioned so that the face of each dentil is half as wide as its height and the cavity of each intersection two thirds of this face in width. The cymatium here is one sixth of the whole height of this part. The corona with its cymatium, but not including the sima, has the height of the middle fascia of the architrave, and the total projection of the corona and dentils should be equal to the height from the frieze to the cymatium at the top of the corona. 
A COMPARISON OF THE IONIC ORDER ACCORDING TO VITRUVIUS WITH ACTUAL EXAMPLES AND WITH VIGNOLA'S ORDER
A: Showing the orders reduced to equal lower diameters. B: Showing the orders to a uniform scale.
12. The height of the tympanum, which is in the pediment, is to be obtained thus: let the front of the corona, from the two ends of its cymatium, be measured off into nine parts, and let one of these parts be set up in the middle at the peak of the tympanum, taking care that it is perpendicular to the entablature and the neckings of the columns. The coronae over the tympanum are to be made of equal size with the coronae under it, not including the simae. Above the coronae are the simae (in Greek ἐπαιετίδες), which should be made one eighth higher than the height of the coronae. The acroteria at the corners have the height of the centre of the tympanum, and those in the middle are one eighth part higher than those at the corners.
13. All the members which are to be above the capitals of the columns, that is, architraves, friezes, coronae, tympana, gables, and acroteria, should be inclined to the front a twelfth part of their own height, for the reason that when we stand in front of them, if two lines are drawn from the eye, one reaching to the bottom of the building and the other to the top, that which reaches to the top will be the longer. Hence, as the line of sight to the upper part is the longer, it makes that part look as if it were leaning back. But when the members are inclined to the front, as described above, they will seem to the beholder to be plumb and perpendicular.
14. Each column should have twenty-four flutes, channelled out in such a way that if a carpenter's square be placed in the hollow of a flute and turned, the arm will touch the corners of the fillets on the right and left, and the tip of the square may keep touching some point in the concave surface as it moves through it. The breadth of the flutes is to be equivalent to the enlargement in the middle of a column, which will be found in the figure.
15. In the simae which are over the coronae on the sides of the temple, lion's heads are to be carved and arranged at intervals thus: First one head is marked out directly over the axis of each column, and then the others are arranged at equal distances apart, and so that there shall be one at the middle of every rooftiling. Those that are over the columns should have holes bored through them to the gutter which receives the rain-water from the tiles, but those between them should be solid. Thus the mass of water that falls by way of the tiles into the gutter will not be thrown down along the intercolumniations nor drench people who are passing through them, while the lion's heads that are over the columns will appear to be vomiting as they discharge streams of water from their mouths.
In this book I have written as clearly as I could on the arrangements of Ionic temples. In the next I shall explain the proportions of Doric and Corinthian temples.