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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