Yup. The Challenges to Squareness.

You didn’t actually think this was going to be easy did you? Well, it isn’t always, but I think the endeavor of overcoming some of these challenges can make things better, especially if you like taking good to great. Knowing what some of the issues can be, and how to overcome them when and if they arise, can help our results better match our desires.

We learned in The Constructs of Squareness article that geometrically speaking, a right angle is 90 degrees, and if it isn’t 90, then it isn’t a right angle. Everything can be represented perfectly on paper, in CAD drawings and in theory, but in building, milling, and manufacturing there are a number of factors, which can affect the quality of accuracy. Some we have to accept, some we can learn to work with, and knowing the difference is how we approach closer to fine, if fine is the goal.

Things that affect the accuracy we use to build do vary. Goals, philosophy, materials and tooling all play a part.

Goals affecting accuracy are often production oriented, cost oriented, or what the intended use of a final product is. If the Goal is to build a doghouse, it needs done quickly, and the price of materials and labor needs kept low, then, the accuracy of squareness need only be relative. If the goal is to make a jewelry box, where scale is small and appearances will be highly scrutinized, then the accuracy of squareness becomes much more important, because the philosophy behind jewelry boxes is seeing how far craftsmanship can be taken. Close tolerance fit and finish is a very large part of how this type of work will be evaluated.

Philosophy does not always have to do with goals, but is often a party to goal-oriented work. All craftspeople over time develop an inner guide regarding the level of accuracy and craftsmanship that are acceptable for their work. Sometimes it is based on the kinds of work they most enjoy, the styles they work in, the level of patience and time they have to give towards their efforts, and if they are working to requirements which are or are not their own.

Materials are often a factor. Consider many different materials, and the methods that render them into a finished product. Casting, molding, rolling, extruding, machining cutting, grinding, all leave behind a surface quality which can affect accuracy, the very accuracy that may be needed to reach a goal. The layout work for a piece of rolled or ground steel may have a smooth surface and take place on a granite surface plate. This smoothness of the surface qualities are enhancements to accuracy.

Wood smoothness is variable, and dependent upon the state of milling it is in. Cutting marks on an 8/4 board from the hardwood dealer can easily be in the +/- .005 to .015 range, and some board surfaces can be found that are coarser than that. Saw tooth marks, planing snipe and other machining factors are the norm. It is up to us to render woods smoother with our own processes, and some woods are rendered smooth from machining processes easier than others.

Moisture content can also play a role. Beyond the limits, which the cellular structure of the wood itself inherently provides, the smoothness from our milling often determines how well we can do with the quality of accuracy we can render upon it. We can mark it for squareness anytime we like, but the quality of squareness we get, no matter how good the tool can be degraded or enhanced by the quality of the surface we are working with. This is why it is a good shop practice to sneak up on the final sizing you need as the board is milled to final dimensions, the process can become more accurate as you go.

Tooling can be a factor. Tooling is available in a number of levels of quality, and accuracy. The higher the quality, of course the higher the cost, and the level of accuracy is commonly better, yet it does not mean that lower quality tools can not be found to be, or made to be highly accurate. It should be evaluated case by case. There are budgets to consider but I’d like to advocate that when it comes to layout tools and approaches to Metrology, it never hurts to do the best you can, and buy the best tool you can afford.

Straightness is a factor, which I touched on, in an earlier article, called The Utility of the Straightedge. Too, squareness benefits from this same straightness, and angular precision is also brought into the mix. Consider a Starrett combination square. It is adjustable, blades are interchangeable on it, commonly to 24 inches, but 36 and 48-inch blades are obtainable. At 24-48 inches away from the squares head, one can begin to realize the value of having straight edges, and precision angular accuracy coming from the tool very easily. If a square with this capability were to contain error, imagine how amplified the error would become at three to four feet from the reference edge.

High accuracy in the tooling pays you. Even when wood surface quality is poor, the layout cannot be more accurate by any tool that is not accurate. The surface quality of the wood can be improved though accurate machine setups, various cutting, planing and machining methods to enhance layout accuracy, if the layout tool can “bring it” to begin with. This means finer accuracy from layout tooling is possible if layout is performed after the surface quality of wood is improved.

Certainly the doghouse we talked about earlier will not be rendered higher quality by using a Starrett precision square, but the jewelry box will suffer if the square used to lay it out was not accurate. If you choose to tool up well, then you are free to work at any level, choosing the level of accuracy you desire, and often even verify the quality of other tools you may own, so you can be aware of the quality of layout they offer, and you can account for, compensate, or restrict the tool for use where it is adequate for the work it is called upon to do.

Wood movement is often at issue, as a reason precision accuracy is not necessary, or desirable for woodworking. Most often, the argument stems from not knowing the ways which wood moves more specifically, therefore ruling out wood altogether as a material capable of high accuracy and precision. Yet those who endeavor to understand wood movement achieve very high quality, stable results from wood in as built conditions.

Here are a few notable thoughts regarding wood movement.

Select grain structure is important. If wood stability for a project is desirable, consider that quartersawn woods are more stable than plainsawn, because the board grain does not cross the pith;

Quartersawn wood orients the growth rings radially, that is, at 45 to 90 degrees to the wide surface. Wood movement is along the rings, and rings are kept short by half or greater in quartersawn boards than that of plainsawn, and the movement from moisture content expected from quartersawn is half that of plainsawn.

Plainsawn lumber deals with growth rings from zero to 45 degrees to the flat side of the board, otherwise called tangentially, it does cross the pith, and movement in these boards can be expected to be twice that of quartersawn.

Vertical, clear grain boards will be most predictable. Avoid boards with interlocked grains, variable grains, reaction woods, tension woods, and mineral deposits, as seasonal movement from moisture content in boards like these is not predictable even at equilibrium moisture. If the figure and beauty these woods can offer is desired, it is up to the woodworker to design with this in mind, and build into their project the compensations needed for these factors.

Once EMC, Equilibrium Moisture Content is achieved, and considering the common ranges of humidity for the area of the country, as well as where the wood will reside in regards to climate control or not, and whether the air handling has air conditioning which is capable of dehumidifying are other factors.

The good news is, wood movement in a climate-controlled area, such as indoors, often has very predictable movement in select, uniform grained boards. A great deal of research and observation has been conducted over the last century, dealing with the many species of woods that are used as building materials, and the data is freely available for use to the builder. Please see Wood As an Engineering Material, written by the US Department of Agriculture, Forest Products Laboratory as one of the foremost writings on this subject. There is a copy available in the Woodworks Library.

The details can be accurately worked out. Properly sawn, select lumber is also helpful when future predictability is desirable, and most boards will not shrink more than 0.2% longitudinally from green to kiln dry, so most any angle cut on the end of a board will remain accurate, as originally cut over the service life of the piece, meaning the accuracy of squareness, can be made highly precise, and can be counted on to remain that way.

Learn to familiarize yourself with the various appearances of grain in wood, and know where and when to put it to the best use. While they can be very, very beautiful, boards of any variety containing interlocked grains, variable grains, reaction woods, tension woods, and mineral deposits will not move consistently when the seasonal moisture in these boards swings. Cupping, twist, warp and wind are all plausible factors, which will affect dimensional accuracy in any direction and the best way to deal with this is to bring these boards to EMC and let them move all they want as they acclimatize.

Begin final milling difficult boards by starting a bit bigger than the intended final size. Work your way in, so as to relieve any of the stresses these boards may have, correcting as you go, so that when you have them at the final dimension, and in the realm of the target EMC, you have the best chance of knowing the future outcomes, and then design around the seasonal movement they will still require.

Once the boards have been milled to the flatness, squareness, and dimensions needed, the next step often includes joinery, and adhesives. Joinery inherently enjoys flatness, straightness and squareness as components of it’s fit and finish. The need for close tolerances is relatively high. Glues commonly call for joinery tolerances for squareness and parallelism of .002-.005 inch, for optimum adhesion, and clamping forces will not help you achieve better results from improper milling.

If the factors affecting wood movement and surface smoothness are observed, wood should be able to be worked and milled easily to accuracy approaching .001 inch, and certainly .001-.003 inch, depending on grain smoothness. As previously discussed, not every project will require this precision in every way, and it will be up to the builder to decide what works best for the project. However if the tooling you own cannot bring .001 inch accuracy, cutting tools cannot mill to accurate lines, and the bets for precision are off, whether the project would benefit from fine looking accuracy or not.

The prevalence of the right angle in engineered structure is probably second only to the straight line in order of importance. Engineered structures in wood are often using plane geometry to help describe and document what they are and how to build them.

Much of the way we think about civil engineering, architecture, woodworking, and even some metalworking, call it flat work if you like, is based on previously understood, maybe even taken for granted, notions about geometry.

Every line that goes in a given direction without variance to that direction is straight, all points that lie upon a line, line segment, or ray can be thought of as congruent. At any point on a line, another line, line segment or ray can intersect, begin sharing a common end point, and create an angle.

There are four ways we look at angles… The most basic angle is the right angle, the angle of 90 degrees, which when measured, corresponds to a quarter of the 360 degrees in a circle, or some thing other than a circle that circuitously begins and ends at the same point. The other ways we describe angles are of angles smaller than 90 degrees which are “acute” and angles larger than 90 yet smaller than 180 degrees, which we call obtuse. If the angle is greater than 180 and less than 360 degrees we call it a reflex angle. When working in terms of squareness, we are only concerned with the 90 degree, or right angle.

Classically, a square has four angles and each of those angles is 90 degrees. If we add all four of those angles together, the result is 360. The interesting thing to note here is that in geometry, and fine work, 360 is not acceptably 359 or 361, and considered a fit. It is either square, or not.

Unlike all the other geometric shapes that use right angles, the square has four sides that are of equal length. This gives us two diagonals, which are also equal. When the diagonals are equal, they are equal to 1.41 times the length of the sides, otherwise known as the square root of two, and this value is referred to as Pythagoras’ constant. These diagonals also form the hypotenuse of right triangles, if the sides of said triangles are equal length.

Now, making your head hurt is not what I am trying to do, but you now know that you can check for squareness if the diagonals within the square are equal length. But what if the sides are not equal length? Well if 2, 3, or 4 sides are not equal, then you don’t have a square, and the angles will not be 90 degrees, except in one case, and that is when each pair of opposite sides of the 4 are equal length, yet adjacent sides are not equal length, This too creates square corners, can be checked with equal length diagonals, and Pythagoras’ Theorem is used instead to find the length of the diagonal. and conversely

Did you guess? This squared, non square is called a rectangle.

Square to the builder is simple, it can be the box or the rectangle, but it is most usable as another name for a right angle. Cutting something to square, or squaring something simply means to form an accurate right angle on the end of it.

With that and a tape measure you can square boards, boxes… Power at your fingertips!

The creation of square where there is not square is easy to do, positioned wherever you need it. You need sharp pencil, a ruler and a compass. Follow along with the diagram, hand drawn by the way, just to show that it can be drawn anywhere. Here are the steps:

With the ruler, draw line AB, and make point A on the left end. With the compass point positioned on point A, swing a short arc mark at any radius length you like, roughly off to the right of the intended perpendicular near the 45 degree radian. Pick any spot you like on that small arc line and cross it to mark your RP, or Radius Point.

With the same compass setting, transfer the point of the compass to the RP and starting at point A, draw a circle based on this unaltered radius length.

At the point where the circumference of the circle intersects line AB, establish point B.

Place the pencil at the RP and position the ruler against it. Align the ruler through the RP and point B. Then draw a line through the circle that intersects point B, the RP, and the circumference opposite point B. Establish this new intersection as Point C. This is Line BC.

Position your pencil on point C, and position the ruler against it. Align the ruler to point A, and draw line AC.

Depending on the how and where this is arranged, this creates a right angle every time. Layout lines for square. The process is called Erecting a Perpendicular. Simple, accurate, scalable, uses few tools. Squareness wherever it is needed. You can even draw it upside down and backwards.

Perhaps line AB and AC already forms the edge of a board or panel. That works. If point A is allowed to be the corner, a point B is established along the horizontal, and a Point C is established along the vertical side, and measurements are taken of line AB and AC. Plugging these measurements into Pythagorean theorem, and conversely will give the length of the hypotenuse, and if it does not measure the same, the board or panel is not square.

In any case, the constructs of square have some pretty simple and humble beginnings, and when observed in working, they help things fit. Now we know a bit more of the back-story. The fit and finish of any project is often defining of many things. Squareness often plays it’s part in the mix, and is often what we are striving for.

Precision refers to the amount of dimensional accuracy or incremental refinement used when something is made, and can be attributed to the quality of the layout, workmanship, or machine set up.

Accuracy refers to the confirmation of dimensional tolerances.

Dimensional tolerances differ with the various types of projects a woodworker will commonly undertake. The set up of shop machines and precision hand tools often requires the precision of accuracy to be at the thousandth of an inch level, however most woodworking projects require accuracy at a level which is commonly referred to by fractions, and is often referred to in the 1/32nd (.031) to 1/64th (.016) range.

The quality in our craftsmanship is inherent in our understanding of these constructs, and our personal stake in setting for ourselves, a level of tolerances. These tolerances are the differences between woodworking, and fine woodworking.

It may not seem relevant, but here is an analogy for higher accuracy. A surveyor will set up an optical instrument, and first, sight their back sight. What they are establishing is a couple of different things, but what is important for us to know for this discussion is that the further away the back sight is from the instrument, the higher the precision of accuracy will be when the surveyor makes other measurements that are shorter than the distance between the instrument and the back sight. The practice is sometimes referred to as going long, and is meant to create higher precision.

One of the common things I have heard over the years, is that in woodwork, a high degree of accuracy is not needed, and then there is the ever ubiquitous, “wood moves anyway”. The understanding being overlooked here is that a lot of assumed accuracy is inherent in the process, because it has been manufactured into the tools we buy, as well as a lot of the lumber we purchase, and we take for granted that it is already “there”. Even wood movement is understood and can be compensated for with relatively high accuracy. None of these assumptions fully get us off the hook.

Consider the ruler. Sure, the ruler has the increments we need, the 1/32nd, and the 1/64th… But we rely on the very same precision accuracy at the fractional level to be consistent to the thousandth of an inch, to assure each of those graduations are where they’re supposed to be. Someone in some lab and factory put all that accuracy into our tools. If we want our precision to maintain 1/64th accuracy, it has to be consistently maintained to 1/64th, plus or minus .001-.002, otherwise the eye will be drawn to errors. After the tool has done its part, the rest is up to us.

Unfortunately, not all levels of woodworking accuracy can be assumed. There are some levels that each of us working the tradecrafts are personally responsible for, and things go better when we are mindful of them.

Take for instance, straightness. The layout of lines is many things, but few things in the layout of lines are required to have the precision of accuracy we have come to expect from straightness. From precision straightness, we can evolve precision flatness, and also use precision straightness as a construct of precision squareness. Parallelism is yet another important derivative of straightness. How straight, straight is, is a pretty important matter. It is always best to start from the best we can do, as it will surely be degraded from there.

Think about the tooling we use to create straightness and flatness. It is inherent in the tooling and machinery. It had to get there somehow. We have to accept that the industrial designers, engineers and machinists did their part, and many woodworkers rely on the good graces of a millwright they never met for a lot of built in accuracy, but there is another part, which they left to us.

One of the more important tools a woodworker can own is a good straightedge. You can have them short or long and there are a number of makers offering them, but if you choose only one, a two-foot straight edge offers a lot of well-rounded utility to the Woodworker. Once you have one, what I want to encourage is; the use of it. Sure they are high accuracy, but it isn’t just for hanging on a peg and looking at.

Straightedges in the woodworking shop have a lot of application. They are available in both steel and aluminum, however they all have more utility if they are made from flat bar stock. Steel straightedges are generally made from stress relieved, 01 steel and are hardened. They are precision milled straight and parallel, and often offer accuracy generally to .001 over the length of the tool. The manufacturer will state the accuracy of their tool, sometimes offering a letter of certification as well. They lay flat on their backs for scribing or drawing lines, and stand on their edges for the comparison of surfaces. They are available with or without beveled edges, and with or without graduations for measurement, but these upgrades are not a necessary requirement, and usually add cost. If you can only afford one, it is better to leave measuring to steel rulers and tape measures.

I find the non-beveled, non-graduated types are less expensive and if it is less specialized, then it usually will offer more utility. Another rational is, that the less it costs the more likely a craftsperson will own it, and if you don’t have one, you can’t put it to good use. There is a lot of good use to be had. Longer than the average ruler and better quality ones are thick enough most generally to stand on edge.

For layout work, the straight edge is a heavy, wide tool, which stays where you put it and has a tall side, which is great for the marking of your work. It is very comfortable for use with any pencil, and it really shines while a marking knife is registered against it. It is an excellent way to connect all the straight lines after you have laid them out. It is also a very nice extension for use with the squares you have and will extend the reach of shorter tools when more reach is needed.

To the hand tool user, the straightedge brings a lot of utility. It can be used to verify the soles of hand planes. After you see where the work needs done, you can then lap the soles to correct the issues and verify as you go. Feel free to verify the flatness of your honing equipment. Flattening the workbench with the use of feeler gauges, a straightedge, and marking is a great use of the tool, because the high spots can be found and removed. The flatness of the workbench is a frame of reference for all future work that comes off it.

Is your board, especially when prepped by hand ready to accept the joinery profiles you intend to put in them? The flatness and trueness of boards is crucial for the fit and finish of dovetails. The plowing of slots and grooves such as dados and sliding dovetails, as well as the treatment provided by hollows and rounds are always made to look a lot better on boards that have been properly evaluated as ready by a straightedge. Handwork is a challenging process; why not evaluate the needed quality before moving to the next part of the process? Besides, the evaluation of a freshly jointed board edge, is just a quick quality assurance check, and a savior before you find an error in mid glue up.

The straightedge is also useful when evaluating the cup, twist, and wind in boards as well as evaluating the flatness of panel surfaces. A pair can even be used as winding sticks. Another good use is for establishing the straightness of the chute edge and fences on a shooting board as well as the overall flatness of its surfaces. While you are at it, evaluate your other shop built jigs from time to time as well.

For machine setups, routine adjustments and maintenance, the straightedge is a great tool. It is invaluable for evaluating the surfaces of the jointer beds for parallel and coplanarity as well as the proper calibration of its vernier settings.

The table saw can be evaluated for table flatness, which is not uncommonly found to be less than perfect yet in some cases correctable. There is also the adjustment of side tables, out feed tables and the trueness of miter slots. It is also valuable to know what the relative flatness and straightness the fence faces have. If there are anomalies, you can then compensate or adjust for them.

Miter saws can use the straightedge for evaluating the trueness of the fence, and are also aided by the straightedge when side wings, when used, are leveled with the main surface of the saw.

A straightedge can also be used for the routine set up of roller stands when used as an in feed or out feed support on any shop machine.

The router table is a high precision shop machine which is commonly shop made. There are many uses for the straightedge with this tool. Evaluation of the tabletop is a constant need with some designs due to the weight the tabletop supports. Many designs are under built and table sag is an error inducing issue. The plates often used to fit the router to the table can be ill fitting in their mortise, and require fine adjustments be made, in order to be brought flush with the table surface.

The router table fence is often in need of straightedge evaluation as well. It needs to be flat and straight, if split, it also has a need for coplanarity. It also must be evaluated to determine if it has any tendency for deflection. The router fence is also a candidate for using a straightedge along with 1-2-3 blocks, gauge blocks and feeler gauges for the settings of the fence and router bit height. With these tools in use, on a well-made table, one can expect fully repeatable accuracy from a router table to be in the .001 range.

The evaluation of any wood, which has been prepared for milling, is important as well. Any cup, twist, warp or wind is something that will throw off the fit and finish of the simplest joinery, and even make edge treatments like bevels, round-overs and more sophisticated profiles look awful. Further, these evaluations can make a lot of difference as to how safe a milling process may be. Knowing ahead of time saves a lot of needless frustration. There are few tools available to the woodworker which can assure things go right, and evaluate why things go wrong, with more power than a straightedge.

If I thought about it, there is probably much more which could be said about such a simple tool, but this is a reasonable well-rounded look at it. It may seem to be a cost prohibitive tool to some, but after thinking outside the box with me awhile, you see it has so much application, and with its evaluatory prowess, how much money could it save you in error free or error caught woodworking, even over the short run? In my shop, it has more than earned its keep and continues to, as I find that wood is costly, even more so than tools. In fact, around my shop, the straight edge offers more value than many other needed tools, and if you can get your mind around that, one will serve you just as well. It can touch so many aspects of your woodworking, that is, if you give it a chance!

I want to have that little talk with you about, Fractions. Yeah. But the plan is, if all goes well, that it won’t hurt – as much as it did last time. Working in sub inch territory usually involves the use of little buggers. The problem many people have when working with fractions, is that they relate the use of the common fraction to their math education experience when they were in school as children.

Our school systems scared the bejeezus out of everyone by forcing us all to learn a series of mathematical exercises, which evolved around fractions that we would never use again in our entire lifetimes. For many, this often created mental blocks to the entire notion of fractions, even the simple useful ones, because after that harrowing experience, it seemed that nothing pleasant could possibly come from the manipulation of fractions at all. In fact, when people are faced with dealing with fractions, they generally feel some panic along with it. A panic that rates up there with the sound of high speed dental drills and root canals, and it is most likely from their harrowing experience in math class. Folks remember what all the wonky practice of solving mismatched fractions was really like, and relate that it was way, way too similar, and maybe even the diabolical preparation, for diagramming English sentences later on during their high school education.

I hope I can help make this a lot more user friendly!

Fractional arithmetic for measurements within the sub inch territory is much simpler than the nightmare a lot of people conceive it to be, because there is a fixed set of fractions that are in use, and memory tricks which you can learn to make well practiced calculations that you can do in your head. This is the reason why the trades have been slow to discard the use of the inch. The fractions are just too handy for halving, quartering and doubling.

There are just some basic constructs we need to know about fractions. A quick few definitions of terms, and memory tricks for using them quickly, and we can be off and running with this for most linear measurement purposes.

Basically put, a fraction is a division problem, which is meant to deal with components of a unit that are smaller than the unit of one. Within the realm of linear measurement, the constructs used are expressing the halving of the unit from the whole on one end, to what is commonly considered usable with common tools on the other. The tools are made to fit the fractional intervals.

Commonly, the scale by incremental division looks like this:
1, 1/2. 1/4, 1/8, 1/16, 1/32, 1/64, 1/128.

In descending order you have a process of halving, and in ascending order you have a process of doubling. Divide or multiply by 2. So there is a memory trick. Get used to doubling and halving units less than one.

Though the numbers in the fractional divisions seem to become larger, it does not indicate the size of the unit, but rather, the quantity of units, which will fit within an inch, so it is best to think of it inversely. The larger the number gets, the smaller the unit is.

The parts of a fraction are worth touching on for a moment.

1 (the numerator) It is the first number or the above number as written.
– Then there is this dividing line, which signifies the division problem.
2 (the denominator) It is the second number or the below number as written.

The dividing line has a name and the name is different depending on how the fraction is written, but the names of the separator do not help you work with fractions easier. So don’t worry about what to call it. For us, “Slash” and “Dash” are fine.

The denominator signifies the sizes that the divisions of one (1 inch in our discussion) are. The numerator signifies the quantity of those denominated divisions we have.

For most of our purposes in linear measurements, these fractions when used strictly as fractions of an inch, will only be added and subtracted to and from each other.

It is important to accept that 1-inch can be expressed as a fraction. No matter what the denominator may say, if you have any quantity of numerators equal to the denominator, the quantity is equal also to one. Such as 4/4ths, 8/8ths, 32/32nds.

For expressing results when two fractions added together create a numerator, whose quantity exceeds the value set by the denominator, then a whole number, such as 1, 2, or what have you, is generated, and the fractional remainder is then expressed with the whole number. This expression is called a mixed number, and is the final expression of your result. As an example you would express 3/4 + 3/8 as 1-1/8, instead of 9/8.

When expressing fractions, they are best expressed in the reduced or simplest form possible. If the numerator is able to be added to another numerator which will derive an even number, the denominator level you are working with may not need to be referred to as a smaller unit, in fact, when numerators added together resulting in an even number, the denominator can likely be expressed as a reduced or simpler unit. As an example: 1/4 + 1/4 = 2/4 = 1/2, and 3/8 + 3/8 = 6/8 = 3/4. It is considered best practice to express the fraction in its simplest form.

A useful property of numbers, which creates a memory trick that we can use, is that if one of two numbers to be added together is an odd number, it will always result in the sum of the numbers being added together to be an odd number. Interestingly, even numbers when added together will result in an even number, and when any two odd numbers are added together they will also result in an even number. Learning to notice this trick will alert you to when the numerator will result in an odd number. When the fractions numerator is an odd number, the denominator cannot be expressed more simply than the finest size being used amongst the mixed fractions, and you will not likely be able to simplify it beyond that resultant fraction. As an example: 7/32 + 1/8 = 11/32. The trick here is that 1/8 has to be converted to its 32nd equivalent, 4/32, and then you can easily add it to the 7/32. The result is as simple as it can be made, because the numerator is an odd number, and cannot be reduced.

Halving all fractions, which are not part of a mixed number is a pretty simple process. Halve the denominator, (multiply the denominator by 2 as this doubles the fractional division making them smaller by half) the numerator remains the same. The result is always half. For example: 3/4 halved is 3/8. 5/16 halved is 5/32. 7/8 halved is 7/16, and so on.

For mixed numbers it is almost as easy. Convert the mixed number into a pure fraction and multiply the denominator by 2. When finished, convert the fraction back to its simplest form, which includes reverting back to a mixed number if that is the simplest form. For example: 1-7/8 = 15/8ths. 15/8 halved is 15/16ths. This is its simplest expression. 2-9/16 = 41/16ths. 41/16 halved is 41/32nds, which is not a simple fraction, so converted back to a mixed number it becomes 1-9/32nds. Please note again, the numerator numbers in simplest expression form did not change from the original expression. Remember when you see the numerator, the trick is that it will still remain the same but the denominator changes by half, leaving little to think about once you remember the trick.

Fractions by the 128th, from 1/128th to 1/4th, as expressed in simplest form. Observe the patterns: 1/128, 1/64, 3/128, 1/32, 5/128, 3/64, 7/128, 1/16, 9/128, 5/64, 11/128, 3/32, 13/128, 7/64, 15/128, 1/8, 17/128, 9/64, 19/128, 5/32, 21/128, 11/64, 23/128, 3/16, 25/128, 13/64, 27/128, 7/32, 29/128, 15/64, 31/128, 1/4. See the interchangeability of denominators?

And finally the last biggie is that fractions only hit specific decimal locations, so sometimes, in order to get to an increment near the fraction you have, you need to convert to a decimal and work it from there. It is simple. Divide the denominator into the numerator for the decimal equivalent. For instance 7/64ths would be converted by dividing 7 by 64, and the result would be .1094, 3/4 would be .750, and 3/32nds would be .0937. How fine is markup to the 128th of an inch? It is .0078 of an inch. Call that about the width of two whiskers.

Why is the fractional to decimal relationship important? Say you are working on adding a shelf to a cabinet project in 3/4 Baltic Birch. This is nice plywood, commonly available, but the thickness is actually metric. 3/4 is close to 18mm but there is a catch. The decimal equivalents are not exact. 3/4 = .750 and 18mm = .709. This is a 41 thousandth of an inch difference, which will need compensation. Usually the compensation is made by working to the fraction nearest to the metric equivalent, which in this case is 23/32nds, but there is a .010 of an inch remainder. This can be an unacceptable gap in some work, so this too is good to know.

Another workaround which creates a better looking fit, could be to disregard the metric size for joinery altogether, and create a 1/2 inch tongue and groove, but in order to center the .500’s of an inch tongue on the .708 inch thick board, you are going to have to subtract .500 from .708, and halve that result of .208 to make it .104, so you know what the shoulders for the tongue will need to measure, in order to center it on the metric board. The nearest fractional increment to the proper size of this shoulder is 7/64 and as you see by dividing 64 into 7 that .109 will make the shoulder too large. The shoulder too large will make the tongue too small. If you cut to the nearest fractional increment here, the tongue will be centered but only .490 wide, and this is a sloppy fit in a .500 groove for very fine work. In fact the same quantity of slop you had working 18mm into 23/32nds. Now you can dial in the necessary compensation.

Knowing how to manipulate the fractions and knowing where the fractions lie amongst the decimals will help you build higher quality into your fine woodworking or machine work, where fit and finish is everything.

Hopefully these memory tricks and conventions will help you to work with fractions faster, easier, and more proficiently with more confidence. The understanding of fractions for use in linear measurements is conquerable, and for the most part is kept pretty simple and doable by the constructs involved with them. One could only hope that much of the rest of fractional manipulation could work as easily.

Going forward, feel free to practice these memory tricks, and if you like, add both a fractional and decimal dial caliper to your metrology tool arsenal, they will help you a ton. For further reference, feel free to use the Decimal Equivilents chart I have provided, as well as the other tools available in the Woodworks Reference Library. They are all printable and ready for use in the shop.

The way measurement is handled in the United States, and to some degree the UK and Canada, depending on the person’s age, is the foot. The foot has an interesting history, and there are a couple different accounts you can go with, but it has its beginnings in the Roman Empire.

Before the world was very big and there was not so much need to measure great distances, measurements were based on what a man had, er, handy! Sure there was mans foot, which is the foot’s namesake, but it didn’t keep a consistent length, so three hands, four palms and twelve thumbs worked better to more consistently derive it. So the Foot became the distance of 12 thumbs, and the width of the thumb became the inch. Welcome to base 12 measurement.

After the fall of the Roman Empire when the Anglo Saxons conquered Western Europe and called their new land, Angle Land (read England) they used the length of 3 barleycorns, as measured from the middle of the ear, placed end to end as an inch, and 6.5 inches was called the shaftment, which is equal to two palms. (Roughly 3 inches)

And so it was, until the Normans conquered England in 1066, whereupon they brought the old Roman system back to usage. King Henry I set the foot at 12 inches, the shaftment at 6 inches, and the yard at 36 inches. The standards for the inch? Three Barleycorns. So we have 36 barleycorn to the foot, and 108 to the yard. King Henry’s standard was made official by an engraving of one foot on the base of a column on the old St. Paul’s Church sometime during his reign. And so using the barleycorn and such, the system ascends upward through the inch the shaftment, foot, the yard, the furlong, the mile… It was known as “by the foot St. Paul’s”

John Barleycorn Must Die.

After the French revolution in the 18th century, the French Academy of Sciences divided the Prime Meridian into quadrants, which is 1/4th the distance around the earth, longitudinally. They then said that 1/ten millionth of this distance will be known as one meter. Going forward since, the want for the most accurate meter possible has become something measured by a standard of light waves in a vacuum traveling one meter as a function of how long it takes.

Descending from the meter we have the centimeter, which is 1/100th of a meter, and the millimeter, which is 1/1000th of a meter.

Because one quadrant, 1/4th the diameter of the earth can be considered such a consistent value, Science quickly adopted the metric system as the definitive method to measure, because all descending units were derived from mathematical divisions of something huge, which offers a real basis for accuracy, as opposed to the foot, which is a derivative of hands, palms, and thumbs, which are defined by three barleycorns.

In 1921 the American Standards Association responded to a request set forth by the Netherlands that a conversion factor be agreed upon between the inch and millimeters. 25.4 was recommended and between attempts by the Ford Motor Company and ASA to get this settled 10 years later, the final installment came in 1959 when the US sealed the deal with the Commonwealth of Nations. The US has been trying to adopt the Metric system ever since.

Since the Metric Conversion Act of 1975, and in many ways prior to that The US Military and many manufacturing firms have gone with the metric system. Despite the mile markers on our highways, many states go with the federal want for the metric system and the roads are built to metric specs.

But is John Barleycorn dead? No. Because the wood manufacturing trades find the good old inch convenient, in it’s fractional ways which can be halved and quartered and such, with the eye being able to see and mark to the 1/64th on average, as well as the way the foot continues to fit well the sizes which Americans like and prefer.

So raise your glasses in auld lang syne to the inch and John Barleycorn. Because ground into your whiskey and your ales, John laughs last, and lives on.

“They’ve hired men with the crab-tree sticks,
To cut him skin from bone,
And the miller he has served him worse than that,
For he’s ground him between two stones.

And little Sir John and the nut-brown bowl,
And he’s brandy in the glass,
And little Sir John and the nut-brown bowl,
Proved the strongest man at last.

Metrology is defined as the science of measurement. More particularly for the woodworker or the home shop machinist/toolmaker, one of the divisions of metrology, which is of particular interest, is applied or industrial metrology. This is about the application of measurement, the suitability of measuring instruments, their calibration, and the quality of the measurements they produce.

So the accurate instrument is applied to create a needed measurement. The quality of the measurements becomes the layout that evolves into successful production. The gist of it is that the woodworker is trying to produce a thing, and the thing is often rendered from a drawing and plans which include materials and cut list. The go between that takes the project off the prints and puts it on the materials being used are the tools of metrology. The measurement and layout tools.

The heavy hitter in the woodworker’s and home shop machinist’s shop then becomes not only the quality of the layout tools, but the knowledge of how to apply them, and the quality of the results rendered by them, and that trifecta is the system of dimensional metrology.

One of the things I have enjoyed down through the years is the process of laying out the work. It is something I have enjoyed professionally, and something I enjoy as a hobby, because the challenge, which comes from, it is always new. I am sure that for some, the problem-solving component of woodworking is what keeps their hand in it. It is not as much the end for some, as it is the means.

In any case, the subject of metrology, most specifically ‘dimensional metrology’, the tools and the layout strategies that they help employ are something of a fun puzzle for me, and will be a part of what I’ll write about here. It has it’s own category, and I’d like to extend the invite to click into the subject from time to time, just to see if there is something there for you.

Layout work is a tedious and exacting part of woodworking. We select boards for size and grain orientation. We hope this is in part, the “art” of our work that separates our project from that which is good, to that of greatness.

We sharpen our tools and skills, we buy accurate measuring and marking tools all with the hope of accurately conveying our vision. We go to work and accurately lay out the work, checking, and double-checking everything as we go to assure we have everything right.

We cut our wood, taking care to get every cut right and of course they are perfect, except for the fact that the face of the board we loved so much is now going to be the back, because we got turned around while we were trying to be so careful. We did not mark our boards properly to affirm their proper orientations in the project above all else.

Can you imagine being half way through with some half blinds when you discover the orientation error?

Oh Man…

A bummer, but it is avoidable.

Buy yourself some chalk and mark your boards as to the intended orientation. Chalk is a buck for a box and it could save you thousands of dollars in layout errors over time. Wipes off, leaves no trace, keeps you on the intended track

Blue painter’s masking tape and a Sharpie marker is a great adhesive notepad for on board paper brain purposes as well.

As an upgrade to plain chalk, consider getting a chalk holder for your piece of chalk. It keeps you cleaner and helps keep the chalk from breaking and rolling off onto the floor. Keep it in the pocket of your apron really nice, you know, where you may find it and maybe remember to use it!

You can also use the blackboard chalk as a release agent on files so that hardwoods, brass and aluminum do not so easily clog the grooves of the file, and this helps make the files work better, last longer and dull more slowly.