Diminution & Entasis

The terms diminution and entasis are used in this context (as stated earlier) for the reduction in diameter from the bottom of the column to the top, and for the curve applied to that reduction, respectively.

Diminution

Regarding diminution, this is an area that is covered in detail by numerous authorities, with quite a bit of disagreement. I have found four different treatments in my research:

Diminution based on column height regardless of the order (following Vitruvius)
Diminution where the Tuscan (and sometimes Doric) orders are treated differently than the remaining orders (such as Vignola)
Diminution where each order is treated differently (such as by Scamozzi)
and Diminution where all orders are treated the same (such as by Chambers).

For this work, the diminution I use will be based on the relevant authority and will be mentioned in the instructions when dealing with each order.

One point I want to raise is that diminution governs the Entablature where it terminates at a corner, as the lowest point of the architrave and the projection of the frieze both align with the upper diameter of the column. The spacing between Columns (referred to as Intercolumniation, and briefly discussed in the section Intercolumniation & Superimposition in the Conclusion) governs the width of the Entablature between Column centers, but the width of the Entablature from the center of the last Column to it’s end is governed by Diminution, and that can affect the placement of recurring elements of the Entablature such as Dentils or Modillions. Therefore, if you plan on using different Diminutions than an Authority specifies with a specific Entablature, it is useful to see whether the combinations will work, or whether adaptations need to be made to one or the other.

Entasis

As stated above, entasis is the curve applied to the diminished diameter of the column, and as this is applied to all columns, I feel it necessary to cover it here, to familiarize you with it before tackling the construction of the individual orders.

Vitruvius mentions this process in his treatise, bud did not provide any more information, so as a result various authorities came up with their own methods of forming this curve.

There are four different methods of forming entasis that I have found and will illustrate below. That of:

Sebastiano Serlio (c.1537)
Giacomo Barozzi da Vignola (c.1562)
Andrea Palladio (c.1570)
and, William Chambers (c.1759).

Placeholder for an illustration comparing the different curves of entasis.

Of the above methods, all of them diminish upwards from a thickness no greater than the lower diameter of the column, except for that of Vignola, whose method diminishes both upwards and downwards from a thickness slightly greater than the lower diameter, at about one-third of the way up the shaft.

Also, of those that diminish only upwards, all of the methods start at a point one-third of the way up the column shaft, leaving the lower part completely vertical, except for that of Chambers, which is applied to the entire space between where the upper and lower cinctures curve into the shaft.

Other than Vignola’s method, the differences between the others is not that great, especially between that of Serlio and Palladio. If you create a pair of profiles using these two methods, they create curves that are extremely close to each other along the entire length. This is true whether you apply entasis to just the upper two-thirds or the entire length of the shaft, and whether applied to the Tuscan or Corinthian columns (or any in between).

Comparing the above two methods with that of Chambers, however, does show a difference, though not too greatly. If you create a profile using the method after Chambers (applied either to the upper two-thirds or the entire length) and compare it to either of the above, you will see that it actually produces a curve that is slightly shallower than the other two. Whether this is desirable or not, is a matter of preference that you need to decide for yourself.

A Note On The Location For Applying Entasis

The location of the start and end points for applying entasis are related to the location of the Upper and Lower Diameters, and whether you are applying entasis to the entire shaft or just the upper two-thirds of the shaft.

Throughout this text I have considered the two diameters to be located at the inner endpoints of the curve of the congé moldings that connect the cinctures to the curve of the shaft itself. This determination I have based on the locations as shown in drawings related to diminution & entasis by Sebastiano Serlio, Andrea Palladio, and William Chambers, as well as by Antoine Desgodets in his measurements of the Pantheon in Les Edifices Antiques de Rome.

Therefore, if applying entasis to the entire shaft (as William Chambers does), you would use the inner endpoints of the two congé moldings as the start and stop points for applying the curve of entasis.

However, during the Renaissance, the more common area for applying entasis was only to the upper two-thirds of the column shaft. Therefore, if you want to apply entasis to only that area, follow the steps below.

Setting The Start & Stop Points For Entasis

This will set the area for entasis to only occupy the upper two-thirds of the column shaft. If you want entasis applied to the entire shaft, then skip this step.

Start by setting a pair of horizontal guides at the inner endpoints of the congé moldings at the top and bottom of your shaft
Draw a construction line, somewhat to the right and separate from your existing geometry, between the two guides mentioned above, divide it into three parts, and set a guide at the top of the first part up from the bottom
Draw a line, starting from the top endpoint of the congé at the bottom of the column, going up to the guide set above
Erase the construction line, but leave the three guides located along the shaft (for the two congé endpoints and the top of the vertical line)
Now use the top of the new line as the start point for applying entasis and the bottom of the congé at the top of the column for the end point

Methods for Applying Entasis

The methods below are shown in order of the date I believe they appeared, not in order of preference or complexity. Which you prefer you will have to decide for yourself, but I can say that the method after Palladio is definitely the easiest to use with SketchUp, while those of Vignola and Chambers are the most complex.

To practice the instructions below, start in a separate file with no other geometry, and start in parallel projection, front view. Using the same units as before (i.e. minutes), create a trapezoidal form, whose left edge is on the Centerline, with a base that is 30 min wide, a top that is 25 min wide, and a height that is 489 min, with the top and bottom representing the the start and stop points for entasis. Once your example geometry is created, place it in a group or component, so it does not interfere with the steps outlined below.

The above example will replicate the area of the Ionic column shaft between the congé moldings, and should serve as a good test for following the instructions below. If you prefer, however, you can create forms that match the heights of the different column shafts, to see how the curves appear for the different orders.

Apply Entasis According to Sebastiano Serlio

This method is described in the chapter on the Tuscan order in Book IV of his Treatise. It was later used by many other authorities, including Vignola, who advised using it for the Tuscan and Doric orders.

Starting with the Column.Shaft.profile visible, with vertical guides marking the Upper and Lower Diameters
Set horizontal guides at the top and bottom start points for entasis
Draw a quarter-circle arc, whose center is the intersection of the guide at the bottom start point for entasis and the centerline of the column, going from the start point for entasis upwards counter-clockwise to the centerline of the column
Where this arc passes over the guide for the upper diameter, draw a small line along this guide so it breaks the arc into two parts, and then erase the larger arc (to the left of the upper diameter) along with the small line
Divide the remaining arc into the desired number of parts (Serlio uses 7 parts)

Draw a construction line, along the Blue Axis a short distance to the right of your profile, spanning between the guides for the top and bottom start points for entasis, and then divide that line into the same number of parts that you divided the arc into above
Set guides at each of the division points on the arc and line just divided, vertically and horizontally
Select all of the divided arc and make it a group (so it does not interfere with the finished geometry)
From the bottom of the divided arc (at the start point for entasis), draw a line upwards and inwards to the intersection of the nearest guides above and inwards from this point
From the previous point continue drawing lines up to the next nearest intersection above and inwards for each division you have, until you reach the bottom of the congé of the upper cincture

Erase any construction geometry (including the group), along with the guides created for this process
Select all the lines, cut them into memory, open your profile component, Paste-in-Place, and redraw an existing line to create the face for your component, then close it

Apply Entasis after Jacopo Barozzi da Vignola

This method is one of two shown in his treatise, the other being that of Serlio. This one he advised using for the Ionic, Corinthian and Composite orders, presenting Serlio’s method for use with the Tuscan and Doric orders. However, I have adapted them so they can be used with any order, and with any Lower Cincture projection.

Starting with the Column.Shaft.profile visible, with vertical guides marking the Upper and Lower Diameters
Set guides at the inner endpoints of the congés at the upper & lower cinctures
Draw a construction line (or vertical scale) between the two guides at the congés, some distance to the right of the Lower Diameter, and divide the line into 12 parts

Set guides at every division of the line
Draw another construction line (or entasis midpoint), starting on the Centerline at a height equal to the top of the 4th division from the bottom of the vertical construction line, going leftwards on the Red Axis for a distance approximately 1 1/2 times the length of the column shaft (a point along this will serve as the center-point from which the angled lines used in forming the curve will rotate)
Set a guide at the projection of the Lower Cincture
Draw another construction line (or entasis scale), starting from the top of the lower congé, leftwards to the Centerline (so it equals the semidiameter of the Lower Diameter)
From the right endpoint of the entasis scale, draw another line out rightwards to the projection of the Lower Cincture, and divide that smaller line into 3 parts, then erase the last part on the right (along with the guide there), and triple-click this geometry and make it a group

Move the entasis scale just created upwards to the top of the shaft and then leftwards a little so it’s right endpoint is on the bottom of the congé of the upper cincture
Draw a circle (with the number of sides increased to 48), whose center is the intersection of the right endpoint of the scale and bottom of the congé, and whose radius is the left endpoint of the scale
Now set a guide, from the center of the circle, downwards & leftwards to the intersection of the circle and the Centerline of the column, making sure this guide continues downwards & leftwards till it reaches the entasis midpoint line
Now set a series of guides, along a path from the intersection of the angled guide set above and the entasis midpoint line, outwards to the intersection of the Centerline and each of the horizontal guides set using the vertical scale

Erase the first, very top, angled guide, the circle and the entasis midpoint line (though leave the guide at the latter location)
Move the entasis scale downwards, so it’s leftmost endpoint is even with the intersection of the first angled guide and horizontal guide set from the vertical scale
Then move/copy the entasis scale down to each intersection of angled and horizontal guides below, along with placing a copy at entasis midpoint guide (which will be the point of maximum entasis, 1/3 of the way up the height of the column shaft)

Now select each scale and rotate it so it is aligned with the angled guide, keeping it’s leftmost endpoint on the Centerline of the column

Select all of the construction geometry created so far and make it a group, and erase the guide marking the Lower Diameter (the guide is erased to make the next step easier)
Now draw a line, starting from the top of the congé of the lower cincture, upwards to the right endpoint of the first entasis scale above it, then continue the line upwards to the endpoints of each of the scales above the previous, till it reaches the bottom of the congé of the upper cincture

With all the lines drawn, you can cut/copy the lines just drawn into memory, erase the construction geometry, open your component, Paste-in-Place, and redraw one of the small lines to create your face

Apply Entasis After Andrea Palladio

This method is one I created myself as an electronic variation of that described by Andrea Palladio in his treatise.

Palladio writes, in the First Book, Chapter XIII, The method I use to make the profile of the swellings is this; I divide the fust [shaft] of the column into three equal parts, and leave the lower part perpendicular; to the side of the extremity of which I apply the edge of a thin rule, of the same length, or a little longer than the column, and bend that part which reaches from the third part upwards, until the end touches the point of the diminution of the upper part of the column under the collarino. I then mark as that curve directs, which gives the column a kind of swelling in the middle, and makes it project very gracefully.

Starting with the Column.Shaft.profile visible, with vertical guides marking the Upper and Lower Diameters, and a stand-in diagonal line occupying the space where your entasis curve will be created
Draw a temporary line going straight down on the Blue Axis from the top of the line marking the start of entasis (if applying entasis only to the upper two-thirds of the shaft) or from the top of the lower congé (if applying entasis to the entire shaft)
Draw an arc, from the top endpoint of the temporary line created above, up to the congé of the upper cincture, and set the curve such that SketchUp’s Inference system reports the arc is Tangent at Vertex to the temporary line and Lower Diameter
Erase the temporary line
Cut the arc into memory, open your profile component, erase the stand-in diagonal line, Paste-in-Place, and redraw an existing line to restore the face for your component, then close it

Apply Entasis After William Chambers

This method is based on the description and drawing in William Chambers’ treatise.

In the chapter Of the Orders of Architecture in General he gives a description of the device he created to apply entasis, which was based on the mechanical instrument described by Francois Blondel (in his Résolution des quatre principaux problèmes d'Architecture, or Solving four primary problems in architecture, of 1673). This instrument itself was based on the method of Vignola.

Starting with the Column.Shaft.profile visible, with vertical guides marking the Upper and Lower Diameters
Set horizontal guides at the top and bottom start points for entasis
Draw a construction line between the two guides marking entasis, some distance to the right of the Lower Diameter, and divide the line into 16 parts, and set horizontal guides at each division

Draw a line, going from the top of the lower entasis start point on the Lower Diameter, inwards to the Centerline on the Red Axis (the length of which should equal the Semidiameter or 30 min)
Draw another line, starting from the bottom of the congé of the Upper Cincture (or upper entasis start point), inwards towards (and past) the Centerline, for 30 min (to equal the Semidiameter)
Draw a circle (with sides doubled from 24 to 48 for greater accuracy), whose center is at the upper entasis start point and with a radius equal to the left endpoint of the just drawn
Now set a guide, from the center of the circle, downwards & leftwards to the intersection of the circle and the Centerline of the column, making sure this guide continues downwards & leftwards till it reaches the horizontal guide for the lower start of entasis
Erase the circle just drawn, along with the line whose endpoint formed the center of the circle
Move/copy the 30 min line at the bottom of the shaft upwards on the Blue Axis so a copy is located at each of the horizontal guides between the two entasis start points, then erase the lowest or original line
Now set a series of guides, along a path from the intersection of the angled guide set above and the guide marking the lower entasis start point, upwards and outwards to the intersection of the Centerline and each of the lines moved/copied above

Now, select each of the 30 min lines and rotate it so it is aligned with the angled guide, keeping it’s leftmost endpoint on the Centerline of the column
Select all of the construction geometry created so far and make it a group, and erase the guide marking the Lower Diameter (the guide is erased to make the next step easier)

Now draw a line, starting from the top of the lower entasis start point, going upwards to the right endpoint of the first rotated line above it, then continue the line upwards to the endpoints of each of the rotated lines above the previous, till it reaches the bottom of the congé of the upper cincture

Erase any construction geometry (including the group), along with the guides created for this process
Select all the lines, cut them into memory, open your profile component, Paste-in-Place, and redraw an existing line to create the face for your component, then close it

A Note on Diminution and Entasis for Pilasters

In the text above, I have assumed you are applying diminution and entasis to columns, but what about pilasters? Regarding them, I have found no complete explanation, so want to touch on the issues here.

In the primary example for creating pilasters in this book, I apply diminution, but not entasis, to the pilasters.

Chambers goes into quite a bit of detail about the issue of diminishing pilasters, but ends up saying that there is precedent both for diminishing as well as leaving them undiminished, though he stresses Nevertheless it is certain that diminished pilasters are on many accounts much preferable.

What he doesn’t say, however, is whether they should be diminished both in depth as well as in width.

Perrault (who agrees with Chambers on this in general) adds that when they are in line with columns we must give the pilaster the same diminution as the column, but this applies only to its front face, since the sides are left without diminution, as they are on the Temples of Antoninus and of Faustina. What he refers to is described by Desgodets: The Pilaster on the outward face does not diminish like the Columns, being full two modules in width at the fillet immediately under the Capital, but the side which faced the Columns was diminished like the columns, in order to support duly the Architrave which passed over both the Columns and the Pilasters.

Therefore, in the primary examples in this book I assume that diminution is applied to the pilaster only on the front view (by which I mean when looked at from the front, the sides will diminish upwards, but when looked at from the side, the pilaster will project an equal amount from the wall all the way up the shaft).

Regarding the projection of the pilaster out from the wall, here I follow Chambers, who remarks they should project one quarter of their diameter beyond the walls, as Scamozzi teaches.

Regarding entasis, I can’t find any references to entasis ever being applied to pilasters. Whether this is a fact, or just my lack of finding sources, I can’t say. However, in the main instructions I will assume entasis is not applied, and therefore the instructions will include modifying the existing column shaft profiles to remove it.

And, finally, there is the issue of where the above will be applied to the pilasters. With the columns themselves there is the option of applying diminution and entasis either to the whole shaft or only the upper two-thirds of it. However, I can’t see how the latter could work with pilasters if you do not apply entasis, as that would create an angle where the straight lower third met the straight upper two-thirds. Therefore, here I assume that if the pilaster has diminution but not entasis applied, that it is applied to the entire shaft, not just the upper two-thirds.