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CONSIDERATIONS & MATH FOR EXTERNAL
ADJUSTABLE RIFLESCOPES
By Wayne McLerran
Posted 3/27/20

Contents:
        Introduction
        Cost Considerations
        Scope Length & Mount Spacing
        Understanding Minutes-of-Angle (MOA)
        Mount Adjustment Range
        Mount Spacing
        Mounting a Scope for Long Range Creedmoor Matches
        Sliding Versus Rigid Scope Mounting
        Parallax Adjustment
        Hermetically Sealed & Nitrogen Filled
        Scope Magnification (Power), NRA BPCR & 22 BPCRA Scope Limits

Having made a significant error in attempting to correct a member
offering scope adjustment advice on one of the well-known firearm
forums, I realized I needed to reeducate myself on the basic
mathematics for installing and adjusting external adjustable scopes.  
At the very least document the information in one place for easy
future access.  Don’t let the title of this article scare you if math
wasn’t one of your favorite subjects in school.  The most complicated
math I’ll touch on is a simple algebra equation.  In addition, I thought
this would be a good time to cover the decision process for those that
may be considering an external adjustable scope.  But before
discussing the physical considerations and mathematics let’s touch on
the typical cost of a scope and mounts.
By the way, it’s common for external adjustable scopes to be referred
to as Wm Malcolm or Malcolm-style scopes after the well-known 19th
century scope designer.  I tend to use the terms interchangeably.

Cost Considerations:
Prior to Montana Vintage Arms’ relatively recent introduction of their
Winchester “B” Series Scope with No.2 mounts for $650 plus shipping,
modern new versions of external adjustable high-quality scopes with
mounts typically cost around $1,000 to $1,200.  Used vintage scopes
manufactured in the 1920’s to 1950’s can be found in good condition
for around $450 to $600.  Another lower cost option is a Leatherwood
scope with either J.W. Fecker or Unertl non-click or de-clicked
mounts.  The mounts that come with the Leatherwood scope are not
suitable for BPCR silhouette or long range competition.  By the way,
non-click mounts are required for NRA BPCR or 22 BPCRA sanctioned
silhouette matches.

Mounts are usually provide with the scope or purchased from the same
supplier. Although mounts used for hunting and recreational shooting
must be of good quality and reliably hold the scope in place,
adjustment repeatability is not a major requirement.  The mounts are
generally adjusted for one distance (typically 100 yards), for zero
crosswinds and not moved.  Black powder cartridge rifle (BPCR)
silhouette competitors must contend with target distances from 200
meters to 500 meters under varying crosswinds and other dynamic
conditions and Creedmoor match contenders are attempting to hit
targets out to 1000 yards.  The rear mount is adjusted many times
throughout matches and adjustment resolution and repeatability are
of the utmost importance.  Therefore, the mounts, especially the rear
mount, must be of the highest quality with excellent repeatability.  Do
not scrimp on the mounts.  If purchased separately expect to pay as
much for the mounts as for the scope.

Scope Length & Mount Spacing:
One of the first likely questions is how long should the scope be?  It’s
certainly an aesthetic decision to some degree, i.e., what looks good
on the rifle, but what a shooter may not realize is the length of the
scope determines the spacing of the mounts, directly affecting the
maximum adjustment amount and adjustment resolution.

Not only is mount spacing determined to some extent by the length of
the scope but is also influenced by other factors.  Mounts for the
relatively long Malcolm-style scopes are generally spaced much wider
than modern-style internal adjustable scopes.  The most common
accepted spacing for shorter external adjustable scopes is 7.2” center-
to-center.  Longer scopes require longer spacing for adequate
support.  As the scope length increases past 23” or so, and depending
on the types of mount used, another commonly used spacing is
10.34”.  For longer scopes, a spacing of approximately 17” may be
necessary.  The reason for the very specific spacing dimensions should
become clearer during the following discussions.

One practical way of looking at the effect of mount spacing is to
consider the scope tube as a lever that pivots on the front mount.  As
the distance from the rear adjustable mount to the front mount is
increased, more adjustment is required by the rear mount to move the
front (objective lens) of the scope and the crosshairs the same

amount.  And this applies to both the vertical (elevation) and
horizontal (windage) adjustments.  Grab a stick or ruler.  Grasp one
end with a hand which will be the adjustment point (equivalent to the
rear adjustable mount).  Now move a finger (the pivot point –
equivalent to the front mount) of the other hand along the bottom of
the ruler away from the adjustment point
while moving the
adjustment point up and down the same amount.  As the pivot point is
moved away from the adjustment point the front end of the ruler
moves less and less.  Hence the movement of the front (objective) lens
of the scope is inversely proportional to the distance between the
mounts.

Since the scope crosshairs are always in alignment with the image
through the objective lens, it should be obvious from the above
example that the total adjustable amount of the image of the
crosshairs on the target (target adjustable amount) is also in inverse
proportion to the spacing between the mounts.  I.e., as the mount
spacing is increased the overall target adjustable amount is
proportionally reduced.  As an example, doubling the mount spacing
decreases the total target adjustable amount by a factor of 2 (cuts it
in half), but it also increases the adjustable resolution by a factor of 2
(doubles it), tradeoffs that must be considered when determining the
length of the scope and mount spacing.  Therefore prior to purchasing
a scope and mounts, it’s important to consider a couple of key factors.
•        Scope length and mount spacing: A longer scope generally
requires a wider spacing between front and rear mounts to properly
support it which, as noted earlier, reduces the total target adjustable
amount.  But a shorter scope can run into problems if the shooting
range is extensive.  For example, attempting to use a 23” or shorter
scope for Creedmoor competition out to 1000 yards may not work
using short equal-height scope bases (blocks).  An example of this
situation is discussed further on.
•        Bullet impact point change: The total rear mount adjustment
range must be sufficient to correct for the amount of bullet drop or
impact point change expected over the shortest to longest target
range.  This is typically measured in MOA (Minutes-Of-Angle) and leads
to the following discussion.

Understanding Minutes-of-Angle (MOA):
A circle contains 360 angular degrees.  Each degree contains 60
angular minutes, also known as minutes-of-angle or MOA.  Therefore a
MOA is 1/60th of a degree.  Although initially it may seem easier for
some to think in terms of inches of bullet impact point change,
experienced shooters and spotters discuss sight corrections in minutes-
of-angle (MOA) to reduce the possibility of ambiguity.  Also shooting
aids such as spotting boards are laid out in MOAs to indicate the
amount of sight adjustment required to correct for off-target hits.  A
simple trigonometry equation is used to determine that adjusting iron
sights or a scope
1 MOA will result in a bullet impact point shift of
1.047” at 100
yards, hence the reason shooters use the rule-of-thumb
of 1” at 100yds, 2” at 200yds and 3” at 300yds, etc.  Some refer to
the approximation as “shooters MOA” vs. “true MOA”.  By the way,
1
MOA at 100
meters is 1.145”.  Regardless, as depicted in the following
illustration, think of MOA as a measurement that gets proportionally
larger with distance.
External ballistic calculators report bullet drop over different
distances in inches and/or MOA.  For an example let’s use a typical
BPCR silhouette match situation: a .45 caliber rifle is sighted in to hit
dead on at 200 meters with a 540gr bullet being fired at a muzzle
velocity of 1250fps.  Using a bullet ballistic coefficient (BC) of 0.350,
Table 1 lists the results of a ballistic calculator out to 500
meters
(for BPCR silhouette) and for longer distances out to 1,000
yards (for
Creedmoor matches).
Note that if the 500
meter drop in inches is divided by the drop in
MOA the result is 5.73”, 5 times the expected value of 1 MOA
(1.145”) at 100
meters.  The same applies for 1,000 yards.  I.e., if
the 1,000yd drop in inches is divided by the drop in MOA the result is
10.47”, 10 times the expected value of 1 MOA (1.047”) at 100
yards.
Mount Adjustment Range:
So now, with a basic understanding of MOA, let’s further discuss the
BPCR silhouette match example in Table 1 in which the maximum
expected bullet drop or impact point change is 51.1 MOA from 200
meters to 500 meters.  Therefore if the shooter plans to only use the
scope for BPCR silhouette the rear scope mount should be designed
with at least 60 MOA of adjustment.  Fortunately most scope mounts
have an adjustment range of 100 MOA or more to easily handle the
above example with front and rear scope blocks of the same height.  
But if the shooter wanted to use the same setup for BPCR silhouette
and for Creedmoor matches the mount options are somewhat limited
and additional factors come into play and must be considered.  More
on this further on.  By the way, keep in mind that the adjustment
range also depends on the mount spacing as discussed in the earlier
stick or ruler analogy.

Mount Spacing:
To determine the best mount spacing considering the above factors
brings us to some more mathematics, the sight adjustment formula
which works for both iron sights and scopes.  Rather than bore you
or complicate this discussion with how it’s derived, the standard
equation is: sight radius (spacing) = rear sight adjustment x distance
to target ÷ bullet impact change.  But since this article is only about
scopes, I’ve replaced the term “sight” in the equation description
with “mount” and also converted it to use MOA values.  Therefore
mount spacing (MS) is equal to the rear mount adjustment (RMA)
multiplied by 3600 and divided by the maximum expected bullet
impact point change in MOA, or "MS=RMA×3600÷MOA".  MS and RMA
values are in inches.  3600 is due to there being 3600” in 100yds,
the bases for MOA as defined earlier.  And since the formula is based
on using MOA it works for any distance.

Using the earlier example, let’s assume the rear mount scale
adjustable range is 0.200” and is marked off in increments of
0.001”.  Using 60 MOA and 0.200” in the equation results in a mount
spacing of 12.0” which will certainly work for a shorter scope but
may not be ideal for other reasons discussed later.  To determine
the minimum bullet impact point adjustment amount, rearrange the
equation as "MOA=RMA×3600÷MS".  Using RMA of 0.001” and MS of
12.0”; the equation results in a minimum bullet impact point
adjustment of 0.3 MOA.  Therefore, each incremental adjustment of
the rear mount moves the bullet impact point 0.3 MOA.  Now let’s
use the same RMA value of 0.001” but shorten the mount spacing to
7.2”, which was mentioned earlier as a standard mount spacing.  
The result is a minimum bullet impact point adjustment amount of
0.5 MOA, or 1 MOA if the rear mount is adjusted 0.002”.  These are
much easier values to remember and use and is the main reason for
recommending very specific mount spacing’s for different scope
lengths.

The previous discussion and calculation assumes the use of a
“shooters MOA” (1.0” per 100yds) in the sight equation).  If the
“true MOA” value of 1.047” is used the mount spacing is 6.88”.  I
mentioned earlier that other commonly used mount spacing’s are
10.34” and 17”.  These are recommended by Montana Vintage Arms
for their longer scopes and are roughly based on using the “true
MOA” of 1.047”.  Using the sight formula the more accurate values
are 10.32” for a rear sight adjustment of 0.003”/true MOA and
17.2” for a sight adjustment of 0.005”/true MOA.

The above examples discuss setting up the scope and mounts to
handle the elevation adjustments necessary over the expected
target ranges.  Windage adjustment is generally not a major
consideration when mounting a scope, assuming the scope body is
properly aligned with the rifle barrel.  But the same equation
applies.  Adjusting the rear mount 0.002” left or right with a mount
spacing of 7.2” would move the bullet impact point exactly 1 MOA in
either direction.

Mounting a Scope for Long Range Creedmoor Matches:
Montana Vintage Arms’ website mentions that their long range
Creedmoor Mount is not recommended for a 23” scope on a 30” or
longer barrel; further stating that at the adjustment top-of-travel
the scope will be pointed at the barrel, thus obstructing the view.  
So if you’re planning on using an externally adjustable scope out to
Creedmoor ranges (800yds to 1,000yds), be aware of potential
problems when attempting to use a short scope.  The following
example will highlight the problem.

Let’s consider an example of a shooter that has a short scope with a
mount spacing of 7.2” on a .45-70 rifle.  He’s been using it for BPCR
silhouette matches in which the shortest distance is 200 meters.  He’
d like to use the same rifle and scope setup for Creedmoor
competition out to 1,000yds with the same loads in the earlier
example in Table 1.  If the rifle is sighted to hit dead on at 200
meters, the ballistic calculator indicates an additional bullet drop of
143.1 MOA at 1,000 yards.  Using the scope adjustment equation
rearranged as "RMA=MS×MOA÷3600" with a mount spacing (MS) of
7.2”, the rear mount would have to be adjusted up an additional
0.29”.  Knowing from the earlier example that 0.002” of rear sight
adjustment is equal to 1 MOA, another way to verify the necessary
sight adjustment is to multiply 143.1 MOA by 0.002” per MOA which
is approximately 0.29”.  Assuming the rear mount has sufficient
adjustment range, many do not, a short scope around 23” or shorter
will not work in this situation with the relatively short front and rear
scope blocks typically used.  The combination of the short scope
length and the tilt angle (to the barrel) necessary for 1,000yds
shooting would result in the scope image being blurred by the muzzle
end of the barrel.

To confirm the problem I used a J.W. Fecker 20.5” long scope with
7.2” mount spacing on a Browning BPC rifle with a 30” barrel using
standard front and rear scope blocks of the same height.  The front
sight had been removed.  The rear scope mount was adjusted to hit
dead on at 200 meters.  After removing the rear block screws, shims
were inserted under the block.  The 1st detectable blurring of the
image was with shims totaling 0.21”.  With shims totaling 0.29”
significant blurring was evident and would be worse with a longer
barrel.  Therefore higher than normal scope blocks would be
necessary to further separate the scope from the barrel.  Another
solution is to use a longer scope in order to position the front
objective lens close to the muzzle.  With either of the noted
solutions, if the shooter wished to use the same setup for both BPCR
silhouette and Creedmoor distances, a rear mount with extended
adjustment range or switching out different height rear scope blocks
would be necessary.

Sliding Versus Rigid Scope Mounting:
Vintage and modern copies of externally adjustable vintage scopes
are generally designed to slide in the mounts to reduce the effects of
recoil on the optics, internal components and mounts.  After each
shot the scope is manually or automatically returned to “battery”
(the firing position).

For hunting situations the preferred setup is a rigid or fixed mounted
scope with a long eye relief.  With the adrenalin rush and the
excitement of the hunt, there’s a good chance the hunter will forget
or not have time to slide the scope back to battery if a quick follow
up shot is necessary.  If a sliding scope setup is used for hunting, one
option is to install a coil spring assembly around the scope body to
automatically return the scope to battery after each shot.  The coil
spring assembly is a common feature found on vintage externally
adjustable scopes but is not allowed in NRA BPCR or 22 BPCRA
sanctioned silhouette matches.

A hundred rounds or more can be fired in matches and the
competitor likely fires many more in preparation.  Although some
have successfully used rigidly mounted scopes for competition, most
prefer a sliding mounting system utilizing a Pope-style rib (see note
below) or similar setup, especially for rifles chambered for cartridges
such as .45-90 or larger.  But even with a smaller chambering why
subject the scope to the full recoil when one with a Pope-style rib or
other attachments are available to allow the scope to slide?  With
each shot the scope slides forward out of “battery”, thus reducing
the stress on the scope and mounts during long strings of shooting.  
Since a return coil-spring assembly is not an option due to being too
large in diameter per sanctioned match rules, the shooter must
remember to manually return the scope to battery after each shot.
Note - A Pope-style rib is a raised rib typically located on the front,
top or bottom, portion of the scope.  It extends a few inches along
the scope and allows the scope to slide in the mounts due to recoil,
but ensures the scope does not rotate in the mounts by the use of a
slot in the front mount designed to accommodate the rib.

Parallax Adjustment:
Parallax adjustment (see definitions below) is another feature to
consider.  Some high-quality scope suppliers offer it, others may
not.  Those that do not provide parallax adjustment suggest that
parallax is not a factor due to the construction and relatively low
power (generally 8-power or less) of the scopes.  Their justification
is that parallax error is minimal with the small diameter and long
focal length lenses used in external adjustable scopes and the
resulting small exit pupil diameters (see definitions below).  Some
competitors are convinced that parallax adjustment offers an
advantage and the setting is changed for each target distance.  But
most BPCR silhouette shooters with parallax adjustable scopes set
the parallax for one target distance (e.g., turkey silhouettes at 385
meters) and don’t change it during silhouette matches, further
justifying the decision by a scope supplier not to offer parallax
adjustment.  If the scope is being used for silhouette and Creedmoor
competition, adjustable parallax should be considered a requirement.
Notes:
Parallax: The displacement or difference in the apparent position of
an object viewed along two different lines of sight.  Without parallax
adjustment or if the parallax is not adjusted properly, depending on
the distance the scope is focused, the crosshairs will move slightly in
relationship to the target in the sight image when the user moves
his/her head.
Exit pupil diameter: The diameter of the image projected on the eye
pupil.  It can be roughly calculated by dividing the diameter of the
objective lens by the power of the scope.  The pupil diameter size of
a human eye is typically 5 to 9mm for young individuals, and
decreases slowly with age.

Hermetically Sealed & Nitrogen Filled:
Due to wide temperature variations and the possibility of wet
weather, a hunting scope should be hermetically sealed and nitrogen-
filled if possible.  Depending on the design and other features
offered such as parallax adjustment, waterproofing an externally
adjustable scope can increase the cost of construction and is
generally not required for most competitive conditions.  
Leatherwood is the only brand of externally adjustable scopes I’m
aware of that are sealed and filled with nitrogen, and parallax
adjustment is not available.

Scope Magnification (Power), NRA BPCR & 22 BPCRA Scope Limits:
This article would not be complete without a discussion on scope
magnification and the additional specifications imposed by the NRA
and 22 Black Powder Cartridge Rifle Association (22 BPCRA) for
“sanctioned” matches.  But first let me say that I’m strictly a big
bore BPCR and 22 BPCRA silhouette shooter.  As of this writing I’ve
launched well over 8,000 big bore and many more .22LR bullets
down range.  Having never shot in BP Target Rifle or Creedmoor
matches I can’t speak for the scope power requirements of either
sport but after discussing it with several shooters the requirements
are the same.

The typical scopes used in big bore BPCR or 22 BPCRA silhouette have
5X (5 power) to 10X (10 power) magnification.  I have used 6X, 8X
and 10X scopes and my preference is 10X when shooting at the big
bore turkey (385 meters) and ram (500 meter) silhouettes.  But 10X
is a little strong for off-hand shooting at the 200 meter chicken
silhouettes for which 5X or 6X would be a better choice.  By strong I’
m referring to the movement of the crosshairs on the target which is
magnified by the scope power.  Therefore 8X is a good compromise.

As noted below, neither the NRA nor the 22 BPCRA specifies the
scope power or length but the current NRA rules include the
following stipulations:
 1) Maximum rifle weight with scope is 15 pounds.
  2) No length or power limitation.  Scope tube body to be ¾” or
less in diameter and any ocular or objective lenses, adjusting or
assembly rings to be less than 1” in diameter.
  3) No internal adjustments for windage or elevation.
  4) Mounts are to be of a traditional style of the period, and
contain the windage and elevation adjustments for the scope in
either, or both, the front or rear mounts.  No click adjustments in
the mount.  Either dove tail mounting or scope block mounting is
allowed.  

The current NRA Rifle Silhouette Rules handbook further states that,
original scope mounts of either the Cataract or Malcolm style or
variations thereof, or replicas or derivatives of either style, are
allowed provided the replica or derivative conforms to the criteria of
rule 4) above.

The same rules apply for  22 BPCRA competition with the exception
that the ocular or objective lens diameter must be 1" or less
measured from the inside of the lens opening.  And the outside
diameters of the adjusting or assembly rings are not included in the
measurements.

For much more information on currently available scopes and
additional
details, see my article titled, Searching for a Cost
Effective BPCR Scope
Solution for Silhouette Competition. Hopefully
the above discussion has provided some insight into the features and
selection criteria when considering or purchasing an external
adjustable riflescope.

Wishing you great shooting.
Wayne