using maths in making bows

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Steven J
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Re: using maths in making bows

#61 Post by Steven J » Fri Aug 01, 2008 3:57 pm

Dave,

I am loving this :D I have gone through page one again and worked a spreadsheet for all the calculations stated. It has been some tough reading but as others have said the maths involved is all about substitution. In reality you have actually done all the work for us :!:

I think that I had better run through what you have said so far and make a bow. I need a little 30lb for some kids at school so I might start there. At 30lb and shooting on our school ABA range they need all the performance they can get from a low poundage bow.

I have attached my spreadsheet for anyone to use or check. It contains the calculations from page 1 of this post. Someone please check this so I know that I am in the right ball park.

Steve
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Re: using maths in making bows

#62 Post by Dennis La Varenne » Fri Aug 01, 2008 4:40 pm

Steve,

No, there isn't any other data on the bows other than what Elmer speaks about in his book unfortunately.

I am inclined to take him at his word in this instance. He was not a boastful kind of bloke in his writings and he was not talking entirely about bows of his own. He speaks of bows of this calibre as being exceptional but not uncommon in his day among the better archers who could afford very good bows. These bows got a pretty heavy workout during comps and had to be able to stand up to a lot of shooting and not let down over the course of a comp, so they would have been made as best as bowyery was capable in those days.

Commonly, the best bows seemed to be made from very good Pacific Yew built to designs by Nagler principally. Elmer's book has a table of bow dimensions for draw weight from the 20+lbers up to the high 60s which Nagler published based on his method. They are all fairly narrow flatbows of rectangular section. I will scan a copy of the table and post it here for everyone.

So far as arrow weights for these bows are concerned, there is no mention, suffice to say that at the level of archery he was talking about, it would be reasonable to assume that the arrows would have been as light as spine would allow in bows shot well out of centre-shot. I doubt they would have gone far into the +400 grains if that, and in good arrows, probably in the mid-high 300s based on my knowledge of the mechanics of arrow spine.

However, he mentions that these little bows were averagely only about 1 1/4" at the flares and 3/8" at the nocks with handles only 5/8" wide and 1 1/2" deep.

In some experimenting I did when a member of a local archery club years ago, I found that my corner-of-the-mouth anchor gave me a point on aim of 45 -50m from my old Perks Staghorn glassed longbow of 50lbs@26" using weighed 500gn arrows with 5 inch parabolic fletches and no overhanging tip (which was used as the draw check). This was on a FITA range.

If I dropped my anchor by one finger's thickness lower on my face, I got near as damn-all to another 10m point-on. Under the chin, it wasn't hard to get a point on at somewhere around 80m plus or minus a bit from my 26" draw. At 28" I could have done better if my arms were longer.

I recently sold a glass lam longbow drawing 60lbs at 28" to a bloke in another FITA club which shot his carbons from 27" (again no overhang) clear over the top by something like the depth of the butt from his under the chin anchor at 80m. FITA longbow has a maximum distance of 70m I believe. We were trying to find the maximum point on distance he could use the bow at, hoping that it would be somewhere on the target face at 70m. At 90m he was still hitting the top of the butt with a point on hold. I don't know how much his carbons weighed though.

Both my old Perks Staghorn and the above bow had a small amount of reflex - about 1".

So, in all, on the basis of the above, I am inclined to give Elmer the benefit of the doubt on this even though the above bows were modern glassed bows. I would be surprised if your very well made glass bows did not do just as well or better. When I get a handle on Dave's material above, I intend to give the method a thorough go.

I am pretty much persuaded that Elmer's claimed level of performance from wood bows is quite possible with the apparent results which Dan Perry talks about in his chapter in TBB vol. 4 on flight bow shooting.

Like all proofs of puddings, it is in the eating of course.

Dennis La Varenne
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Re: using maths in making bows

#63 Post by yeoman » Fri Aug 01, 2008 7:30 pm

To add to what Dennis has said, there are bowyers in the US who make very fast selfbows.

Tim Baker, of TBB fame made an 84 inch longbow from a $10 board that outshot fita bows for speed. Cool stuff. After this thread is through, or maybe by the end of this thread, I'll share a similar bow but shorter, with 3mm wide tips.

Looseplucker, I humbly thank you for your kind comments.

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Re: using maths in making bows

#64 Post by Dennis La Varenne » Sat Aug 02, 2008 12:15 am

Everyone,

Attached is an extract from Elmer leading up to Nagler's tables about which I wrote above.

Dennis La Varenne
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Re: using maths in making bows

#65 Post by Steven J » Sat Aug 02, 2008 10:11 pm

Dave,

I have been mucking around drawing some tiller diagrams for bows as you have shown in this tutorial. It is great to be able to see what the tiller will look like before the bow is built - I only hope that I can put this into practice soon.

My question:

Is there an optimum string angle to be achieved? If we define string angle by measuring the angle of the string to the tangent of the curve at the bow nocks, what sort of angle should we be aiming to achieve in our bows. We want to avoid stacking by having too great an angle, but is there a problem with having a bow so long that the angle is small. To much rotational inertia to overcome maybe?

Ed,

Needing a CAD programme for this and other things, I have downloaded CadStd as you suggested. I can't seem to get the hang of the arcs required to draw the bow. Is this a limitation with the programme or me? I have actually found it a pretty tricky programme for the couple of hours I have spent today. It does not feel very friendly to use. Any suggestions???

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Re: using maths in making bows

#66 Post by yeoman » Sun Aug 03, 2008 12:50 pm

Hi steve,

When the bow is pulled from brace, the string has a massive amount of leverage on the tips. As the draw progresses, this leverage becomes less and less. That is, the bow begins to stack.

The least leverage exerted by the string on the bow tips is when the angle between the string and the tangent of the curve is 90 degrees, I believe.

While CADstd can definately be used, I personally use DeltaCAD, which is a bit more user friendly. I can do a short tutorial on it if you like.

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Re: using maths in making bows

#67 Post by looseplucker » Mon Aug 04, 2008 12:27 pm

Aw, shucks Dave - I meant it all too!

I reckon its great that you are sharing all this stuff - its light years ahead of what I can do - but we'll plug away and see what happens.
Are you well informed or is your news limited?

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Re: using maths in making bows

#68 Post by Steven J » Mon Aug 04, 2008 2:32 pm

Thanks for the offer of the tutorial. I have downloaded Delta Cad and another called Draft IT. Draft IT seems good but I can't seem to figure out how to divide an arc into segments without purchasing an upgrade from the free option. Delta Cad also will cost me money after 45 days. Why do I expect everything to come for free these days I will never know.

Anyway if you could offer a quick (and I know that your time is precious) on how to divide the arc in Delta Cad so that I can determine perpendicular distances to the string that would be fantastic.

Steve.
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Re: using maths in making bows

#69 Post by yeoman » Mon Aug 04, 2008 5:48 pm

OK, here's how to split a circle:

Once you've drawn your circle, click on the 'edit' tab.

Click on 'split line or circle' button. It is a straight line with a dot on it, towards the right next to the 'Z' with an up arrow next to it.

Click somewhere on the circle.

This splits the circle in two.

Erase half of the circle. Preferably the half you don't need.

Click on the 'slide an endpoint' button, which looks like a mostly upside down '!' mark, just to the right of the erase 'ER' button.

Click on one end of your arc and move it to where you want, then click again to set it in place. Repeat for the other side.

If you want to split this up into sections, I suggest copying the whole picture and pasting it further down the page, so that if you stuff up you can go back to it.

Anyway, in the edit tab still, click on 'E', or 'Edit' function, then click on your arc. A little window will pop up with various information such as length, start point, end point, radius etc.

At the bottom is a 'convert to lines' button. Click it. then enter as many sections as you want (don't go overboard) and click OK. The curve has now been turned into a set of straight lines. Each end of each line rests on the arc of the circle.

Is that what you're after?

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Re: using maths in making bows

#70 Post by Steven J » Mon Aug 04, 2008 10:08 pm

Thanks Dave,

I'll give it a go tomorrow, its getting late. I know that you probably had to go through the motions with DeltaCad to get the instructions down accurately - many thanks.

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Re: using maths in making bows

#71 Post by Steven J » Tue Aug 05, 2008 12:38 pm

Dave,

I got the deltaCad worked out. It is a pretty simple programme to use. I encourage others to have a go.

My next problem is with my bend tests. The figures that I seem to be getting for stress seem to be in the wrong order of magnitude. I thought that I should ensure that I am getting the same figures in my spreadsheet using your data before plugging in my own. I can't replicate your calculations.

When I run the spreadsheet using your data for the example posted on July 31 at 5:32pm I cannot arrive at the same figures that you end up with.

My MoE is 20649 and yours is 14000
My stress is 12.22 and yours is 117

The data you posted was
Load (kg)-----deflection (mm)--------set (mm)
9------------------21-----------------------0
10-----------------22.5--------------------0
11------------------24.9-------------------0
12------------------27---------------------2
Can you check my spreadsheet and let me know where I am wrong. Have I just missed something simple somewhere?

Steve
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Spreadsheet for Dave's Calculations.xls
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Re: using maths in making bows

#72 Post by longbow steve » Tue Aug 05, 2008 6:46 pm

Thanks for the reply Dennis, the more I think about it the more feasible it becomes to have a point on distance at 100 yds considering draw lengths and he may have had a long face :D no doubt exceptional performance from an allwood bow with inferior string materials etc. Thanks also for the reading material.

Dave, was Tim's 84" olympic recurve devouring l'bow referred to in The Bowyers Bibles or can it be found on Paleo Planet forum?
Thanks Steve

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Re: using maths in making bows

#73 Post by yeoman » Wed Aug 06, 2008 6:07 pm

Well Steve, I've had a look at your homework, and I think I have a few solutions for you. And me.

The first is rounding. Where you use a figure of 9.8 for the force conversion, I use 9.80665. While this doesn't seem much, when you multiply it to the order that we do in these formulas, the difference becomes quite large. Large anough to be many megapascals out.

But that's not the main issue.

the biggest issue is, in fact, mine. Yep, I MADE A MISTAKE. the MoE value that I gave was not in fact correct. 14000 was the number which I found in the spreadsheet, and neglected to check it. the real MoE, according to my calculations, is 20662 or therabouts, which is very close to your calculation above. This mistake of mine is a grave shortcoming of my own neglect, and to all those who have read this and become confused, I apologise to the three of you.

I will edit my previous posts so as to contain the correct figures.

The bending moment is a simple matter of mistaken identity. It isn't, in fact (force x deflection)/4, but rather (force x length)/4. this may be another mistake that I made int he transcribing of the formulas into this thread. I'll check and adjust if necessary.

So it is that the fault is not your own, you who faithfully followed my instructions. Rather, it is my fault for not double or triple checking my post before I submit it, and for that I'm sorry.

I've adjusted your spreadsheet...corrected it, so that it now reads the same as mine, which in turn has been checked by engineers who do this sort of thing for a living but with bridges and skyscrapers and whatnot.
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corrected spreadsheet for Steve.xls
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Re: using maths in making bows

#74 Post by Steven J » Thu Aug 07, 2008 8:35 am

Many thanks Dave,

No apology necessary for the mistake made. Whilst you are leading the way, I figure that we are all working on this together. Often the only way that we notice mistakes is by having someone questioning us so that we examine our methods thoroughly. Thats what I try to instill in the the students that I teach (Highschool - Science, Chemistry and Agriculture).

Anyway, I seem to have the theory down pat now. I can do a bend test, draw the bow diagram in Delta Cad, and figure the limb thickness and widths. Now all I need to do is put it into practice and build myself a pyramid bow. :D

Would you be able to provide a list of references where you sourced your knowledge?

Have a good day.

Steve
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Re: using maths in making bows

#75 Post by yeoman » Sat Aug 09, 2008 11:19 am

Well,

most of the material on how to design and calculate the dimensions of a bow I got from Archery: the Technical Side, which was a part of the Legends of the Longbow series.

The bend test information I got from a few engineers via pers comms.

If you teach science, maybe this sort of project, from bend test to bending bow, would be a good 'real world' application of scientific/mathematical rinciples for the students to do. :?:

Dave
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Re: using maths in making bows

#76 Post by yeoman » Fri Aug 15, 2008 6:44 am

How're we all going with this?

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Re: using maths in making bows

#77 Post by GrahameA » Fri Aug 15, 2008 7:25 am

Slowly

I really should sit down and apply some of it but I seem be a tad busy at the moment. :D
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Re: using maths in making bows

#78 Post by Brett Finger » Fri Aug 15, 2008 8:03 am

I got lost back at the push bike annolgey... :roll:

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Re: using maths in making bows

#79 Post by Steven J » Fri Aug 15, 2008 8:12 am

I'm pretty comfortable with the theory and have done many drawings and calculations. I am going to start a pyramid bow this weekend to test it all out. I'll let you know how it all goes.

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Re: using maths in making bows

#80 Post by Dennis La Varenne » Fri Aug 15, 2008 3:14 pm

Dave,

Just keep it coming and don't wait to see how we are going with it. That's up to us.

You are the tutor on this. We can ask questions when we are stumped.

The beauty of this kind of tute is that it is there for good and we can always go back to it to check that we are on the right track. It is also good that posting any queries lets others see if they are having similar problems.

I am still digesting a lot of it, but I will commence a bow shortly too.

You have already solved on of my principal problems on how to allow for optimum stress on a limb at varying drawlengths when you posted that -

S = draw force / cos angle.

Using Hickman, I was using a very convoluted method of making this simple allowance.

Dennis La Varenne
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Re: using maths in making bows

#81 Post by Steven J » Fri Aug 15, 2008 3:29 pm

Dennis,

I missed the point on calculating the optimal stress from the formula you mention. Is the draw force you in the formula force in newtons or pounds?

I was just looking up Archery: The Technical Side when the new reply to post message came to the email.

Dave, you have certainly inspired me.

Steve
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Re: using maths in making bows

#82 Post by yeoman » Sat Aug 16, 2008 3:27 pm

Steven, I think he means the diea of calculating optimum stress at different draw lengths due to the fact that the stress calculations are based off the tension in the string, which is a geometrically deduced figure, hence the S=force/cos angle.

The tension (s) should be in Newtons.

Dave
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Re: using maths in making bows

#83 Post by Steven J » Sun Aug 17, 2008 10:15 pm

Dave,

Didn't quite understand your explanation of Dennis' formula, but I am happy to listen to further explanation when you have the time. :D

I have begun a bow by thickness sanding a piece of Grey Ironbark to the thickness required by the calculations and drawings that I have made. It is not the best way but I am looking for the easy way out for the moment while I experiment. I have taken the slat down to thickness and have glued on a riser piece, however I know that I may be inviting troubles with the fades begining at a glue joint. It is quite possible that the handle my pop off, but also possible that it may hold and survive.

You have mentioned that Nagler's method enables us to see that the limb tips should be zero width. This is obviously impossible, and even more unrealistic when you consider that extending past the tips to allow for a string nock requires a negative wood value (anti-wood?). :?

How do you practically blend the theoretical design and the practical requirement to have a position for the string? TBB4 discusses the 'Eiffel Tower' shape, however TBB1 and 2 discuss bows that are made with pyramid limbs that having straight sides that rather than the slightly swelled sides that Nagler's formula helps us design.

Is it of more value to the practical bowyer to use Nagler to determine the width at the fades and run the limbs parallel to the tips and tiller as per normal to obtain the arc of circle tiller?

Anyway, if I can take five minutes and stop building my laminated bows I might get to answer these questions for myself.

Steve
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Re: using maths in making bows

#84 Post by Dennis La Varenne » Tue Aug 19, 2008 7:44 pm

Dave and Steven,
Steven, I think he means the diea of calculating optimum stress at different draw lengths due to the fact that the stress calculations are based off the tension in the string, which is a geometrically deduced figure, hence the S=force/cos angle.

The tension (s) should be in Newtons.
Dave has it exactly. I should have indicated that S stands for stress in Newtons as Dave picked up.

Steven,
With regard to built-up handles, the old timers overcame the problem of handles popping off by leaving the middle of the bow raised and rigid with a dead flat platform on which the handle block was glued. The dip from the glued on handle was faded into the limb well below the glue line of the handle which kept the middle of the bow rigid. I hope is clear from the attached diagram. This made a bow which was basically a rigid handle bow with a built up section to form a useful handle.

Regards,

Dennis La Varenne
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Glued-on handle riser.jpg
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Re: using maths in making bows

#85 Post by Steven J » Tue Aug 19, 2008 9:40 pm

Dennis,

The riser glue up you illustrate is typical of the other all wood bows that I have built and makes good sense for the longevity of the bow. I am attempting to take a short cut with this bow (maybe foolishly) to test the maths involved. I figured that it was easier (not better) to thickness sand the whole bow minus the riser and glue the riser block on to the already thicknessed slat. The strength of the epoxy is extreemly high, and I figure if I run down gently into the limbs from the handle the glue should hold. A more abrupt fade may cause problems.

When I build a more study thickness sander than the drill press and drum that I used for this, I will be able to leave a raised portion in the middle of the bow slat.

As for the stress formula, are you saying that the optimal stress for the timber in the limbs is the same as the stress on the string. I thought that the stress was determined through the bend test and looking for an appropriate amount of set. Sorry to cause frustration to you in asking for further explanation.

Steve
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Re: using maths in making bows

#86 Post by Dennis La Varenne » Wed Aug 20, 2008 12:20 am

Steven,

Belay what I said above. I just realised I have got it all arsy turvy. This discussion is about bows of constant limb thickness.

The angle of the string and hence the distance from string to neutral axis governs the width required to ensure a constant and evenly applied 'stress at elastic limit' at all points along the limb for whatever wood is used. If I draw up my scale diagram as Dave has just shown us, but with the string at my 26 inch draw length instead the 28 inches in his diagrams - all my previous problems using Hickman in relation to varying draw lengths are taken care of.

Whatever was I thinking of?????

Dennis La Varenne
Dennis La Varénne

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yeoman
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Re: using maths in making bows

#87 Post by yeoman » Wed Aug 20, 2008 6:39 am

OKay okay gather around everyone, let's do some clearing up.

S=force/cos angle is for determining the TENSION in the string at your intended draw length. S=tension in this case, and is measured in newtons.

A bow can be calculated to experience optimum stress at whatever draw length you choose because of the genius of this formula. With the hickman formula, this option did not exist.

How's that?

Steve, I'd wrap that handle as well. As strong as you can.

Dave
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Steven J
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Re: using maths in making bows

#88 Post by Steven J » Wed Aug 20, 2008 8:13 am

Thanks for the further explanation gentlemen,

As I understand it..

By calculating the tension in the string (from our chosen draw weight and the string angles as given in our drawings), and by knowing the perpendicular distance from the string to the bow limb (also from our drawings), one can calculate the correct limb width at any point for the chosen stress we wish the timber to take (we choose the elastic limit to ensure that we make the most of our timber properties) at any drawlength chosen (again because we geometrically determine distances and angles from our drawings)

If I still have not understood the tension thing, I might at this stage just nod in an unconvincing manner so that we may move on :)

No doubt as I continue with this experimentation and calculation I will come to understand the maths more clearly.

Have a good day,

Steve
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On Christ the solid rock I stand, All other ground is sinking sand. Edward Mote, 1797-1874

Dennis La Varenne
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Re: using maths in making bows

#89 Post by Dennis La Varenne » Wed Aug 20, 2008 7:54 pm

Steven,

After having done your bending tests and worked out the elastic limit of the wood you are using as well as the modulus of elasticity, and the other preliminary calculations,

First there is this -
R/T=E/2S where:
R=radius
T=thickness
E=MoE
2=2
S= stress desired/required in the bow limb
Then there is this -
So, we have our formula, R/T=E/2S but we need to know only the thickness. The rearranged formula is thus T=2RS/E I know that formula almost spells a naughty word for bottom. Mathematician s have fun too you know!

Let’s finally substitute some values.
T=unknown
E=20663
S=137
Thus:
T=2RS/E
T=(2x26x137)/20663
T=0.34 inches
And finally we get onto this -
The formula for figuring the width of a bow’s section is given as:

W=(6dt)/(S(T^2))
Where:
W= width
d= distance to string
t= tension
S= desired stress
T= thickness
So, your surmise that -
As I understand it..

By calculating the tension in the string (from our chosen draw weight and the string angles as given in our drawings), and by knowing the perpendicular distance from the string to the bow limb (also from our drawings), one can calculate the correct limb width at any point for the chosen stress we wish the timber to take (we choose the elastic limit to ensure that we make the most of our timber properties) at any drawlength chosen (again because we geometrically determine distances and angles from our drawings)
is correct.

I think we are both thinking the same way. Dave will let us know if we are off track. (. . . and this is only for constant thickness limbs of circular tiller. Elliptically tillered limbs will be interesting.)

Dennis La Varenne
Dennis La Varénne

Have the courage to argue your beliefs with conviction, but the humility to accept that you may be wrong.

QVIS CVSTODIET IPSOS CVSTODES (Who polices the police?) - DECIMVS IVNIVS IVVENALIS (Juvenal) - Satire VI, lines 347–8

What is the difference between free enterprise capitalism and organised crime?

HOMO LVPVS HOMINIS - Man is his own predator.

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Re: using maths in making bows

#90 Post by Steven J » Wed Aug 20, 2008 10:13 pm

Thanks Dennis. This reminds me of spending a few hours in the library at university after lectures with your mates trying to figure out what Professor Dave was on about. :)
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On Christ the solid rock I stand, All other ground is sinking sand. Edward Mote, 1797-1874

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