Classical Proportioning

The classical orders have almost always been proportioned independent of real world units. This was the method used by Vitruvius, and adopted by all classical theorists since then. Once the proportions have been established, you can then translate those dimensions into the local measuring system of your choice.

There have been a number of different proportioning systems developed over the years, as different authorities tried to discover what they thought were the best and most convenient.

The most common proportioning methods that I am aware of are:

by Equal Parts of Other Parts (as used by Vitruvius, Serlio, and Gibbs)
by Parts of the Module (as used by Vignola)
by Minutes of the Diameter (as used by Palladio and Chambers>
by Large, Mean & Small Modules (as used by Perrault)
by Fractions of the Diameter (as used by Ware)
and by Decimal Fractions of the Diameter (as used by Chitham).

Below I will give more information on each of these systems and how they work, but first I will explain the different ways you can implement proportioning using SketchUp.

Proportioning in SketchUp

Whichever of the above methods of proportioning you choose, they can be used with the steps and techniques in this book in several different ways.

Using Real World Dimensions and a Divided Scale

With this method you would create a scale that is equal to the size of your Lower Diameter, and then use that to proportion your model from.

This is most useful with dimensioning by equal parts, or fractions of the diameter. It is also very amenable for use with dimensioning by parts of the module, or by large, mean & small modules.

Another benefit is you don’t have to worry about what measuring system you use, whether imperial or decimal, and can also adjust the proportions based on the final size (very useful with adjusting proportions for use in furniture).

Using an Independant Unit and Scaling the Result

With this method you would set your lower diameter to equal the unit size of your method of choice, then scale the final result up or down as needed.

This is most useful with dimensioning by minutes of the diameter, as well as parts of the module. On the other hand, it could be very difficult with dimensioning by decimal fractions of the diameter, at least without a workaround to deal with thirds and sixths.

For the purpose of this work, it is this method that will be used, both for convenience and due to the authority I will be following for my example orders below.

Using Translated Unit Dimensions

With this method you would use real world dimensions in your modeling, but instead of a scale in SketchUp that translates proportions you would have the dimensions noted down elsewhere, in a document or spreadsheet, or input ad hoc from a calculator, where the proportions are translated to the specific real-world size of your model.

This is most useful with any of the dimensioning methods except for by equal parts, as there you do not have a single number to use in your calculations, but must calculate in stages.

Using a combination of any of the above

A final method, and one that I often use, is a combination of two or more of the above, where I might use dimensioning by minutes of the diameter, but then subdivide other parts, or where I translate subdivisions or parts into minutes. In this way, I can easily compare the proportions of different authorities and see how they relate.

Also, I don’t believe you have to be held to a single proportioning or scaling method. You can combine them all together, for example, by translating Vignola’s parts into Palladio’s minutes, then subdividing a smaller part by dividing it as desired, and choosing dimensions that are clean fractions of the diameter. This is especially important to keep in mind when designing your own set of proportions for the order of a given project, or adapting an existing set of proportions for a smaller adaptation of an order (like a miniature column on a cabinet, for example).

Examples of the above methods will be provided for each dimensioning method below, and indicating which I think are the easiest or most problematic with each given system.

Methods of Proportioning

By Equal Parts of Other Parts

The method of dimensioning by equal parts is where the whole order, or parts of the order, are subdivided into a number of equal parts, each of which are then subdivided again and again, until the smallest elements have been dimensioned. The starting point was frequently either the lower diameter of the column, or the column as a whole.

This is the process described by Vitruvius in his treatise, and adopted by Serlio and his followers later in the Renaissance. It was later reintroduced, in a more systematic form, by James Gibbs, in the 18th century, who applied the method to the entire order at once, and provided multiple scales on his diagrams to show the proper subdivisions of each element as it relates to the greater whole.

Using real world dimensions and a divided scale to model from is the easiest with this method, as you don’t start off with an independent dimension, just a number representing either the total height of the order or the width of the diameter.

Using an independant module size and scaling the result can easily be done with this method, just use the same process as above, but give the scale the independent number instead of the real one.

Using translated module dimensions from an outside source can be done with this method, but is very time consuming since there is never a single unit to convert by, as well as the fact that, depending on what dimension your lower diameter has, the conversions can get tricky and end up with numbers very hard to input into SketchUp.

An example of this would be that the Attic base is 1/2 the Lower Diameter, it’s plinth is 1/3 of it’s height, the remaining 2/3 is divided into 4 parts, the uppermost of which is the upper torus, the 3 parts left being divided into 2 parts, the lower of which is the lower torus, with the remainder divided into 7 parts, with 1 each for the fillets and the rest for the curve of the scotia (as per Serlio).

This sounds complicated, but, depending on how you draw your scales, it can actually be very simple.

Example: The Doric Attic Base of James Gibbs

the Doric Attic Base of James Gibbs.

The following assumes you are using a Doric order with no pedestal

Draw a line, equal to the entire height of the order, starting on the Origin Point and going upwards on the Blue Axis, then move it to the left a little so it is off the Centerline
Divide this line into 5 parts, and set guides at the top of the 4th and 5th parts, for the heights of the column and entablature
Set guides vertically and horizontally on the Origin Point for the Centerline and Baseline
Draw another line, a little to the left of the first, equal to the height of the column, and divide that line into 8 parts, giving you the size of the Lower Diameter, as the Doric column is 8 diameters high
Set guides a half part down from the top and a half part up from the bottom of this last line, giving the heights of the capital and base
Draw another line, a little to the left of the last, up to the lower guide just set above, giving you the height of the Attic base
Divide this line into 3 parts, and set a guide at the top of the 1st part, for the height of the plinth
Draw a line, a little to the left of the last, starting at the guide just set and going up to the top of the base, and divide this line into 8 parts, setting guides 3 and 5 parts up from the bottom, for the top of the lower torus and top of the scotia curve
Then set guides a half part above each of the two guides just set, for the heights of the lower and upper fillets of the scotia

Select lowest part of the second scale drawn above, which represents the Lower Diameter, move it down and over so it is below the Baseline and aligned on the Centerline, then rotate it 90 degrees counter-clockwise from the top
Divide this line into 2 parts, and set a guide at the end of the first part, which is the projection of the Lower Diameter from the Centerline
The projection of the plinth equals it’s height, which is 1/3 of half the Lower Diameter, so divide the second, outer, part of the above into 3 parts, and erase the 2 outer parts and set a guide at the new endpoint, giving you the projection of the plinth
Divide this last part into 4 parts, and set guides at the midpoints of the 2 inner parts, the inner guide giving the projections of the upper fillet, midpoint of the upper torus and projection of the lower cincture over the base, while the outer guide gives the projection of the lower fillet

At this point, you would then follow the instructions for creating the various molding forms making up the base.

By Parts of the Module

The method of dimensioning by parts of the module is where the order as a whole is divided into 15 equal parts, with 3 for the entablature, and 12 for the column. If there is a pedestal it becomes 19 parts, with the additional 4 being for the pedestal. Then, the column is subdivided based on the specific order used, i.e. 14 parts for the Tuscan, 16 for the Doric, 18 for the Ionic, and 20 for the Corinthian & Composite. It is this last division that produces the module that is used, and which is itself subdivided into 12 parts for the Tuscan & Doric orders and 18 parts for the Ionic, Corinthian & Composite orders. These parts, or fractions thereof (in combination with the module for larger elements) are then used to proportion all elements of the orders.

This system was devised by Vignola, originally for his own use, and is that presented in his treatise Regola delli cinque ordini d’architettura of 1562.

It has been described as radical, with Christof Thoenes on the Les Livres d’Architecture website saying The result is an arithmetical model, and with its help each order, harmoniously proportioned, can easily be adapted to any given height, of a façade or an interior.

Using real world dimensions and a divided scale to model from is quit easy with this method, as once you figure out the size of the module, using the steps above, you can draw a scale of that size and divide it into the appropriate number of parts.

Using an independant module size and scaling the result is very easy with Vignola's method, just set your diameter to 24 or 34 inches or centimeters, and then model using Vignola's dimensions. When you are done modeling, you can resize the entire model or individual components, by measuring a specific distance (like the diameter) and typing in the real-world dimension you want.

Using translated module dimensions from an outside source can be done fairly easily with this method, as you could enter in all of Vignola’s dimensions into a spreadsheet, and then multiply or divide as needed by either 24 or 36 parts or modules.

An example of this would be that the Attic base is 1 module or 18 parts high, the plinth is 6 parts high, the lower torus is 4 1/2 parts, the fillets of the scotia are 1/2 part each, the scotia is 3 parts, and the upper torus is 3 1/2 parts (per Vignola).

Example: The Attic Base of Jacopo Barozzi da Vignola

the Attic Base of Jacopo Barozzi da Vignola.

The following assumes you are using Vignola's scale where 18 parts equal a module or half the Lower Diameter, and that you are forming this for his Ionic order.

Draw a line, equal to the entire height of the order, starting on the Origin Point and going upwards on the Blue Axis, then move it to the left a little so it is off the Centerline
Divide this line into 5 parts, and set guides at the top of the 4th and 5th parts, for the heights of the column and entablature
Set guides vertically and horizontally on the Origin Point for the Centerline and Baseline
Draw another line, a little to the left of the first, equal to the height of the column, and divide that line into 18 parts, giving you the size of the Module, as the Ionic column is 18 modules high and the Lower Diameter is 2 modules wide
Move/copy the lowest part of the last line a little to the left, and divide it into 18 parts

Select this 18 part line and make it a component Vignola.Scale.18th
Set a guide 6 parts up from the Baseline, for the plinth, then another 4 1/2 parts up for the lower torus, 1/2 part up for the lower fillet, 3 parts up for the scotia, 1/2 part up for the upper fillet, the remaining 3 1/2 parts being for the upper torus
Move/copy the Vignola.Scale.18th down and over so it is below the Baseline and aligned on the Centerline, then rotate it 90 degrees counter-clockwise from the top
Set a guide at the end of this scale, for the projection of the Lower Diameter from the Centerline, then move/copy the scale from it’s left endpoint to it’s right, so you can dimension the projections of the base
Set a guide, coming out from the Lower Diameter, for 7 parts, for the projection of the plinth

Set a guide, coming inwards from the greatest projection, at 4 1/2 parts, for the upper fillet, all coming inwards from the greatest projection of the base, for 2 3/4 parts for the lower fillet and upper torus, and 4 1/2 parts for the upper fillet
The next guides have to be set at 3/4 projections, so draw a line, starting 2 parts inward, going down on the Blue Axis a little, then continue it inwards on the Red Axis so it is equal to 1 part of your scale
Divide this new mini-scale into 4 parts, and set a guide 3 of those parts inwards from it’s right endpoint, for a total dimension of 2 3/4 parts from the greatest projection, for the lower fillet and upper torus

At this point, you would then follow the instructions for creating the various molding forms making up the base.

By Minutes of the Diameter

The method of dimensioning by minutes of the diameter is where the diameter is divided into 60 minutes, which are then used to proportion the order.

A distinction must be made here, however, as the moduleused can be either the diameter or half the diameter. As Palladio explains it, The module shall be the diameter of the column at bottom, divided into sixty minutes; except in the Dorick [sic] Order, where the module is but half the diameter of the column, divided into thirty minutes. Chambers simplified this usage, saying I have chosen the simplest, readiest and most accurate, which is by the module, or semi-diameter of the column, taken at the bottom of the shaft : and divided into thirty minutes.

So, while the number of minutes for the lower diameter remained unchanged, the number of minutes in a module depends on the authority and the order being used.

This technique of using 60 minutes to the diameter was first introduced by Palladio, and later adopted by a number of other authorities, including Scamozzi and Chambers.

Using real world dimensions and a divided scale to model from can be done with this method also, using the same technique as for dimensioning by parts of the module, but it is a little harder due to the larger number of divisions you are dealing with.

Using an independant module size and scaling the result is also real easy with this method, and is, in fact, the method that will be used in this book. Here I set my Lower Diameter to 60 inches, and then use Chambers’ dimensions input directly into SketchUp. When I am done, I save my component to my filesystem, and when I import it into a new model I scale it to the size I need.
Using translated module dimensions from an outside source is just as easy with this method as for dimensioning by parts of the module, and uses the same technique, only with 60 minutes.

An example of this would be that the Attic base is 30 minutes high, the plinth is 10 minutes, the lower torus is 7 1/2 minutes, the fillets are each 1 1/4 minutes high, and the scotia and upper torus are each 5 minutes high (per Palladio’s text description).

Dimensioning by Minutes of the Diameter.

By Large, Mean & Small Modules

The method of dimensioning by large, mean & small modules involves using a great module, equaling the diameter of the column, a mean module, that is a semi-diameter (or half of the great module), and a small module, that is a third of the great module. Smaller divisions are then handled as follows: What we usually call a part - the thirtieth part of half a column diameter - will always be called a minute in this treatise in order to avoid the confusion the word part might cause. Here, part does not signify a fixed part, as the word minute does, but rather a relative part, such as the third, the fifth, etc., of another part.

In addition, projections of the parts are determined by dividing the small module into five parts, whose subdivisions are used for the dimensions.

This style of proportioning was shown by the French architect and theorist Claude Perrault in his Ordonnance for the Five Kinds of Columns after the Method of the Ancients of c 1683 and is one I personally find perhaps the most complicated.

Using real world dimensions and a divided scale to model from can be used with this method also, again following the same technique as for dimensioning by parts of the module, only in this case you would be creating five scales: one for each module used, plus one equaling the half diameter for the thirty minutes, and then a final one for dimensioning projections. In addition, there would be the use of temporary scales to divide as needed for small parts.

Using an independant module size and scaling the result can be done here just like with dimensioning by minutes of the diameter, as Perrault actually uses 60 minutes to the diameter for part of his dimensions. I would, however, also create scales to show the other modules he uses.
Using translated module dimensions from an outside source could be done here, similar to dimensioning by either parts or minutes, but, due to the multiple modules used, can, again be somewhat troublesome to figure out what units need to be converted against what dimensions.

The examples I have been using, that of the Attic base, is, in Perrault’s work, divided up using equal parts, just like Serlio’s, with the exception that the scotia and fillets are divided into 6 parts, like Palladio’s description.

Dimensioning by Large, Mean & Small Modules.

By Fractions of the Diameter

The method of dimensioning by fractions of the diameter involves reducing all the basic elements into simple fractions of the lower diameter of the column, with the smallest parts being subdivisions of these fractions.

This way of dimensioning is illustrated by William Ware, in his American Vignola of 1904, where he describes it as I remember well the day when, as I was carefully drawing out a Doric Capital according to the measurements given in my Vignola, Mr. [Richard] Hunt took the pencil out of my hand and, setting aside the whole apparatus of Modules and Minutes, showed me how to divide the height of my Capital into thirds, and those into thirds, and those again into thirds, thus getting the sixths, ninths, eighteenths, twenty-sevenths, and fifty-fourths of a Diameter which the rules required, without employing any larger divisor than two or three. It seemed as if this method, so handy with the Doric Capital, might be applied to other things, and I forthwith set myself to studying the details of all the Orders, and to devising for my own use simple rules for drawing them out.

Using real world dimensions and a divided scale to model from is very easy with this method, though at the same time somewhat laborious, as you would create a diameter-sized scale for each part you needed to dimension, and then move/copy it and divide as needed.

Using an independant module size and scaling the result could be used with this method, but instead of a number associated with this method, you could use the diameters from Vignola or Palladio. This is especially useful if you want to compare proportions from different authors, and/or if you can easily convert in your mind the minutes or parts to fractions of the diameter.
Using translated module dimensions from an outside source can be very easy with this method, as you can simply set your diameter to the real-world unit and then divide or multiply the dimensions as needed.

An example of this would be that the Attic base is 1/2 the Lower Diameter, it’s plinth is 1/6 the diameter, the lower torus is 1/8 the diameter, the fillets are each 1/48 the diameter, and the scotia and upper torus are each 1/12 the diameter (per Palladio’s text description translated into fractions of the diameter).

Dimensioning by Fractions of the Diameter.

By Decimal Fractions of the Diameter

The method of dimensioning by decimal fractions of the diameter is similar to the preceding, in reducing everything to subdivisions of the lower diameter, but here the lower diameter is given a value of one unit, with everything being decimal multiples or fractions of that number.

This is the latest method I know of and was created by Robert Chitham. In describing it in his book The Classical Orders of Architecture of 1985, he writes that he wanted a system amenable to division by ten so as to be sympathetic to the Metric System, and which could easily be used with a pocket calculator. As a result his proportions are given in decimal fractions of the diameter of the column at its base.

Using real world dimensions and a divided scale to model from with this method is, as far as I can see, almost impossible without further help. The trouble is, when dividing a line to create your scale, you have to have a number to divide it by, which is not provided with decimal fractions, unlike imperial fractions. You would need to convert the decimal fraction first, which would defeat the purpose of the scale.
Using an independant module size and scaling the result can be done with this method, but is both very easy and problematic. The issue is you can set your diameter to 1 decimal unit, and directly input numbers from Chitham’s dimensions, but when you try to actually model you run into problems with components and elements being too small, so you have to end up temporarily scaling things up and down as you go in order to get them to work properly.

Using translated module dimensions from an outside source is even easier with this method than the last, as you are using the single unit and decimal numbers to convert from.

An example of this would be that the Attic base is 0.5 units, it’s plinth is 0.16 units, the lower torus is 0.125 units, the fillets are each 0.025 units, the scotia is 0.075 units and the upper torus is 0.09 units (per Robert Chitham).

Dimensioning by Decimal Fractions of the Diameter.

Converting Dimensions

As mentioned earlier, you can convert dimensions from one system to another, but, just as converting from a dimensioning system to real-world units, the conversions can be tricky at times. This is especially true in converting from equal parts and decimal fractions.

If you are converting parts of the module, minutes of the diameter, or fractions of the diameter, they all can be converted to each other fairly easily.

The table below provides some common conversions between the various systems.

Comparison of Proportioning Dimensions
24^ths	36^ths	60^ths	Semi-⌀	Full-⌀	Decimal
1/4		5/8	1/48	1/96
		3/4	1/40	1/80
1/3	1/2	5/6	1/36	1/72
		1	1/30	1/60
1/2		1 1/4	1/24	1/48
		1 1/2	1/20	1/40
2/3	1	1 2/3	1/18	1/36
3/4		1 7/8	1/16	1/32
		2	1/15	1/30
1		2 1/2	1/12	1/24
		3	1/10	1/20	0.05
		3 1/8	5/48
	2	3 1/3	1/9	1/18
1 1/2		3 3/4	1/8	1/16
		4	2/15	1/15
2	3	5	1/6	1/12
		6	1/5	1/10	0.1
2 1/2		6 1/4	5/24
	4	6 2/3	2/9	1/9
3		7 1/2	1/4	1/8	0.125
		8	4/15	2/15
	5	8 1/3	5/18
3 1/2		8 3/4	7/24
4		10	1/3	1/6
4 1/2		11 1/4	3/8	3/16
	7	11 2/3	7/18	7/36
		12		1/5	0.2
5		12 1/2	5/12	5/24
6	9	15	1/2	1/4	0.25
8	12	20	2/3	1/3
12	18	30	1	1/2	0.5
18	27	45	1 1/2	3/4	0.75
		48		4/5	0.8
		50		5/6
		50 3/4		11/13 (or 5 1/2 of 6 1/2)
		52		13/15 (or 6 1/2 of 7 1/2)
		52 1/2		7/8	0.875
		53 1/3		8/9
24	36	60	2	1	1.0

If you are interested in using a specific method of dimensioning your orders, you can adopt the designs of authorities using other methods, and just translate those designs into the dimensioning system of your choice. However, while in theory this is easy, in practice it can provide difficulties. A few examples will suffice.

For example, to convert Vignola’s 24^th Parts to Palladio’s 60^th Minutes, you simply divide 60 by 24, to understand that each of Vignola’s 24^th Parts is equal to 2 1/2 of Palladio’s Minutes. Likewise, each of Vignola’s 36^th Parts is equal to 1/36 of a Diameter (per Ware), while Serlio’s Upper Torus of the Attic Base (specified as 1/4 of 2/3 of 1/2 the Diameter) equals 5 of Palladio’s Minutes (30 divided by 3, then 20 divided by 4).

Similarly, Chitham makes the upper torus of his Attic base 0.09 decimal units. Multiplying that by 60 gets 5.4. But his plinth is 0.16, which, multiplied by 60, comes out as ~9 19/32 in SketchUp (or 9.6 using Windows Calculator).

Note: Conversions can get a little tricky however, as sub-dividing Equal Parts can give some very odd fractions, while Chitham’s Decimal Units have serious issues when dividing into threes.

Which Proportioning Method to Use

The answer to this really depends on a number of factors, including:

Which authority you are following (if any)
Which real-world units you are using for modeling (as some you might feel are more amenable to one type more than another)
Which is easiest for you to remember and understand
and Which is just what you prefer using more than another.

Therefore, the one you choose is up to you.

The only two that am personally not fond of are those of Perrault and Chitham, the former as it just seems to complicated to use three different modules, and the latter as it is difficult to use decimal fractions when dividing by three, which is very common in Classical Architecture (and is a point Chitham himself points out, along with his workaround)

Also, I (personally) tend to think (and model) in more than one proportioning system, frequently taking ideas or examples from one system and using them in another. For example, I might want to compare the works of Serlio, Palladio, Vignola and Chambers, so would model to one scale (say the 60th minutes) but draw a scale to subdivide to see how something compares to what Serlio advises and translate a dimension to see how the other three compare. I do this as I usually model in Palladian minutes, but actually prefer the proportions of Vignola over Palladio, so I also keep thinking in terms of those ‘parts’ translated into ‘minutes’ (i.e 2 1/2 minutes equaling a 24^th part and 1 2/3 minutes equaling a 36^th part). In addition, I keep going back to the methods used by Vitruvius, Serlio, Gibbs, and Ware, as the idea of all the parts being related to each other in a proportional relationship is one I find intellectually intriguing and appealing.

The real answer is what I stated at the beginning, that it is really up to your individual preference. One thing to keep in mind though, is that you are not bound to follow the Dimensioning System of an Authority if you prefer another method to actually construct them. You can always convert the dimensions from one system to another (as noted above with the example by Chambray).

Dimensions & Units

Having spoken of proportions and how they are used to dimension the classical orders, here is some information on how that relates to real-world units as used in this work.

Here are a few more bits of information on dimensions and units of measurement as they are encountered in this book.

I use Architectural Inches as my Units of Measurement in SketchUp, with a precision of 1/64 inch, length snapping set to the same precision, and angle snapping set to 1.0 units. I mainly do this for three reasons:

Many, if not most, classical treatises on the orders use dimensioning systems that employ fractions, which are natural to express in these units
Classical architecture uses multiples of three for many things, including dimensioning, and they are difficult to express accurately in decimal units (whether those be imperial or metric)
and, as an American it is what I am used to and grew up with, so it is most comfortable for me to work with.

However, this does not mean you have to use Architectural Inches, you can instead use Decimal Metrics or Inches. As you will see below, the choice is up to you.

For ease of both translating dimensions and modeling without any issues, I associate one Architectural Inch with one Palladian Minute, so that the Lower Diameter will equal 60 inches, thus making each minute (or min) equal to 1 inch. For those using the Metric system, you can set the Lower Diameter to equal 60 centimeters so that 1 min would equal 1 cm. Once the element has been completed, the component can then be scaled up or down to the desired final dimension. This makes it extremely convenient to model the works of William Chambers, who will be the main source of my examples here.

The dimensions used in classical architecture very frequently involve the use of fractions. Due to the nature of SketchUp’s interface, however, many of these will not appear the same as they are entered. For example, entering “1/3” will appear (when measured by SketchUp) as “~21/64”, and likewise “2/3” will appear as “~43/64”. This is due to the software not measuring by thirds (as even if you lower the precision level from 1/64 down to 1/4 you will end up with “~1/4” and not “~1/3”, for example. This is just something to be aware of as you proceed.

When using decimal units with either the imperial or metric systems, the fractions used for dimensions in the text will have to be translated to their decimal equivalents, which in certain instances can cause conversion errors, as dimensioning by thirds is rather frequent in Classical architecture. When using SketchUp you can input fractions when using decimals, allowing the program to do the conversion for you. If you do not want to adopt this method, however, you can use Robert Chitham as your authority, as his treatise The Classical Orders of Architecture uses decimals for their dimensions, and he provides several ways of dealing with this issue.

All height dimensions will be given for individual elements, measured from the element just above or below, following common practice of classical authorities. Thus, if told to set Guides 2, 3, and 5 min up from the Origin, you would set a Guide 2 min up from the Origin line, then another 3 min up from that point, and the final 5 min up from the last Guide created, for a total height of 10 min. Also, the heights will be listed in the order they appear, going up or down from the modeling starting position.

All projection dimensions (unless otherwise noted) will be measured from the same starting location, not measured sequentially from the last point. Thus if told to set Guides 11, 15, and 18 min out from the Upper Diameter, you would set a Guide 11 min out from the Upper Diameter, another 15 min out from the same starting point, and the last 18 min out from the same starting point again. Also, the projections are listed in the same order they appear for the heights, though this sometimes means that you are setting guides for the greatest projections first.

With this information, hopefully you will be able to follow along with the instructions later in this work with the minimum of trouble or issues. The main thing to keep in mind is that the proportional dimensions used are not necessarily mapped to real-world units; I only do so for ease of modeling. If you prefer to convert the dimensions rather than map them to a real-world unit one-to-one, you may use the methods mentioned above to do so.