At full draw the arrow does not point at the centre of the target, and yet that is where it ends up. On a bow without a centre-shot window, as on many traditional bows, the shaft rests against the side of the grip and its axis runs off to one side of the centre; on a modern recurve the angle is small, but the arrow still does not leave in the direction it points. This is the archer's paradox, and the explanation lies in something that is now well measured: as it leaves the bow the arrow flexes, and the way it flexes depends on its spine as well as on the bow and the release.
The most solid measurements come from footage shot at 4,000 frames per second of sixteen world-class recurve archers. As the string rolls off the fingers it carries the nock towards the archer's body, by 13 millimetres on average; then the shaft straightens, bends the other way, comes away from the button within 8 milliseconds and leaves the string just under 15 milliseconds after the release begins. It passes the riser only afterwards, when it is no longer resting on anything. In the only test that compared the two releases on the same arrows, a mechanical release left the arrow flexing less than a fifth as much as in shots by two archers off the fingers, but it still flexed. How this picture came together, from the first accelerometer mounted on a bow in 1987 to today's models, I describe on the page about the archer's paradox in research; here I look at what those measurements are for when you have to choose and tune an arrow.
What the paradox is
The problem starts with geometry. The arrow has to pass beside the bow, so at full draw its axis does not coincide with the line to the centre of the target: if it flew rigidly in the direction it points, it would land to one side. Instead it reaches the centre, and it is this apparent contradiction that gave the phenomenon the name of paradox.
On a modern recurve, with the sight and the button set up, the misalignment is small but it can be measured. With the bow mounted on a shooting machine and the sight centred on a target at 20 metres, the drawn arrow was aligned about 220 millimetres to the right of the centre, against the 226 predicted by calculation: as it leaves the bow its centre of mass picks up a small sideways velocity to the left, for a right-handed bow, and the arrow lands where it was not pointing. For the compound a model reaches the same conclusion, that for a right-handed archer at full draw the arrow should be aligned a little to the right of centre, even with the rest at zero lateral offset. There, though, it is only a calculation, checked by eye on the target without saying how large the shift is, and in the registry the claim is transferred.
What the arrow does in the first milliseconds
The best-documented sequence is that of the sixteen world-class recurve archers, eight men and eight women, filmed from above at the London Olympic test event. When the archer relaxes the fingers the string does not leave straight: it slides across the fingertips, moves sideways and takes the nock with it towards the body. The largest displacement comes on average 5.4 milliseconds after the release begins and measures 13.1 millimetres, from 9 to 20 depending on the archer. At that moment the shaft is bent in a curve, with its middle pressed against the button; straight afterwards it starts to straighten and comes away from it, in all sixteen within 8 milliseconds, is straight at around 8.6 and then bends the other way. The nock leaves the string on average at 14.7 milliseconds, with the shaft still bent in this second direction.
The fingers start it, but the flex does not come from them. With the bow clamped on a machine and no hand on the string, four types of arrow still flex sideways, by 4.6 to 5.5 millimetres measured from 80 centimetres in front of the bow onwards, and footage of a Polish national-team archer, twenty years earlier, already showed the same shape with the archer and with a shooting machine. That is why in the registry the claim that the arrow flexes and snakes around the riser is corroborated. What pushes the shaft sideways is the way the force reaches it: the string drives it from the nock, the front section is held at the side by the button, and the point, the heaviest part, is the slowest to get moving because of its inertia.
The push along the shaft, in turn, adds to the flex. Engineers call it column loading, and on the shooting line you hear that an arrow that is too weak, compressed like this, buckles sideways like an overloaded column. In the calculations that exist it does not: the only estimate on real equipment, a compound with a 500 spine arrow, gives a peak axial load of about 60 newtons against a critical load of about 100, and in the model of an average target bow the thrust also stays below the threshold. In the registry the idea that the arrow buckles during the launch is therefore contradicted, but by scant and indirect evidence: the margin shrinks with more draw weight, more point weight, and longer or weaker shafts, and nobody has instrumented an arrow during a shot close to the limit.
How much the way the fingers open matters was tested with a Japanese shooting robot fitted with a competition recurve and motorised fingers. The authors expected faster opening to make the arrow flex more, and it did not: with the motor at 30, 90 and 150 rpm the peak flex was 7.07, 7.44 and 7.23 millimetres, highest at the middle speed, and opening the fingers only until the string slipped away brought it down to 6.94, eight hundredths of a millimetre from the archer used as a reference. The differences are a few tenths of a millimetre, on a machine and with no statistical test, so the finding that flex does not grow in proportion to finger speed is reported and nothing more. The frequency at which the shaft vibrated, on the other hand, did not change in any condition.
As for the shape, it is almost always a simple curve that changes direction, the shaft's first bending mode. In the footage of the sixteen archers a small second-mode component is often visible too, giving the shaft a slight S shape especially early in the power stroke, while after the arrow leaves the bow the first mode dominates. In the laboratory the second mode was measured at about three times the frequency of the first, 241 hertz against 80.2; that the arrow also vibrates in higher modes is reported by a single research group, and how much it matters in the shot has not been measured.
Passing the riser
The practical question, for anyone tuning, is whether the arrow touches the bow on the way past. In the same footage the point where the shaft passes closest to the button is not the tail. In the archer whose full sequence the authors show, the nock leaves the string at 14.5 milliseconds, the minimum gap between shaft and button comes a millisecond later, about 150 millimetres from the nock, and the nock passes the button only at 18 milliseconds, with ample room for the fletching too. The arrow passes the riser, then, when it has already left the string. At that point it has only just started to roll about its own axis, so the vanes go past oriented as they were at full draw, which is why nock orientation is chosen to give them as much room as possible.
The clearance is not the same for everyone, though. Across the sixteen archers the gap with which the tail cleared the button ranged from about 2 to 25 millimetres and followed the ratio between the arrow's two flexes, which the authors link to the way the bow hand twists the grip: nine archers who did not seem to twist the bow passed with 8 to 10 millimetres, five with 2 to 5. The twisting is estimated from the images rather than measured, and there is only one study, so the claim is reported; it does suggest, though, that clearance at the riser depends on the hand as well as on the arrow.
For the shaft to pass cleanly, two models by the same author, eight years apart, give the same condition: it has to complete a full cycle of vibration while it is still on the string. In the example from the first, with a modern target bow, the stiffer shaft stays on the string for 16.2 milliseconds against a vibration period of 15.6 and passes without touching; the weaker one stays on for 15.5 milliseconds against a period of 18.9, leaves after three quarters of a cycle and brushes the grip with its tail. This is where spine enters the paradox, because it sets the period at which the shaft vibrates and therefore whether the cycle closes in time.
A 1998 model adds the ideal moment for separation: the string should release the arrow as the nock crosses back over the bow's median plane, because at that instant the string is no longer pushing it sideways. In the footage of the sixteen elite archers, however, only four had the nock leave the string on the median plane or very close to it; for the others the separation point scattered, with a standard deviation of 4.9 millimetres and one case at 15. The difference showed afterwards too: when separation happened away from the plane the string kept swinging sideways, in the case closest to the plane the string stopped almost at once. That only a minority of elite archers separate on the median plane comes from this one study, and nobody has shown that separating there leads to tighter groups or higher scores.
In flight: the arrow vibrates and swings
Once clear of the bow, the arrow makes two different movements, which are often confused on the shooting line. The first is the vibration of the shaft, the same flex as before carrying on. In the footage of the sixteen archers the first bending mode in free flight lay between 80 and 105 hertz, with thirteen arrows out of sixteen between 80 and 87, and in a competition shaft the two still points of this mode fall about a fifth of the length from the nock and 7 centimetres from the point. Within the same series the stiffer shaft vibrates faster: three ProTours, from the 470 to the 380, vibrate at 74, 77 and 80 hertz in one measurement and at 78.3, 80.2 and 83.3 in another by the same research group. The stiffer shafts are also heavier, though, and the step from one spine to the next is worth two or three hertz: that a stiffer spine vibrates at a higher frequency is reported, and the frequency has never been tested as a way of choosing the shaft.
This vibration dies away slowly. The shaft's own damping, measured in two different laboratories, is small: setting a carbon arrow resonating at 84.8 hertz gives a damping time of about a second and a half, longer than the flight at competition distances. That the arrow reaches the target still vibrating is an inference, though, because the calculation leaves out the air, and how far the remaining vibration shifts the point of impact has not been measured.
The second movement is the swing of the whole arrow about its centre of mass, and that is the one the fletching damps out. The centre of pressure sits towards the back, near the vanes, and as soon as the shaft is at an angle to the airflow a moment arises that brings it back into line and sets it swinging. Paper tests at several distances, on arrows shot from a compound mounted on a machine, find one cycle every 19 to 22 metres and a swing that dies down in about 26; in simulations of the recurve the cycle is 14 to 18 metres, and the waver in the trajectory shows in the first twenty metres of seventy. The angles are small, between a quarter and half a degree in still air. How quickly the arrow straightens also depends on the vanes: three different laboratories find that the type of vane changes it, and in the widest comparison, on a model arrow in a wind tunnel, the world's most widely used vane brings it from 45 degrees of incidence back to zero in 0.934 seconds, against 1.283 for an average vane.
Nor is the flex necessarily just a fault to keep down. A Japanese laboratory launched the same arrows in four ways, with two expert recurve archers, a robot with mechanical fingers, a mechanical release and a compressed-air launcher that does not make them flex, and worked out the air drag shot by shot. At the speed archers shoot, mean drag did not change appreciably, but its spread did: the standard deviation of the drag coefficient was 0.073 with the archers, 0.103 with the mechanical fingers, 0.119 with the release and 0.267 with compressed air. The authors conclude that flex of the right amplitude makes the layer of air round the shaft turbulent, and with it the drag more repeatable. It is a single experiment, with few launches per condition and no statistics on the differences: that an arrow shot by an archer has more consistent drag is reported, and whether this shows up as a tighter group on the target has not been checked.
Spine: the number that governs the flex
If flex is the heart of the paradox, spine is the number you choose it by. It is the stiffness of the shaft, how much it resists bending, and together with length it is the first thing to look at when you change arrows.
Static spine is measured on a bench: you rest the shaft on two supports, hang a weight from the middle and see how far it sags. In the ASTM standard used today for carbon and aluminium arrows the span is 28 inches, 711 millimetres, the weight is 1.94 pounds, about 880 grams, and the spine number is the deflection in thousandths of an inch. A ProTour 380, for example, sags by 9.65 millimetres, which is 380 thousandths of an inch.
Which way the number runs often confuses beginners. Since the value is the deflection itself, the more the shaft bends the higher the number: a 500 arrow is softer than a 400, and the big number means the weak spine.
Static and dynamic: why the same arrow changes
Static spine is measured with the arrow at rest, but in the shot the shaft is driven by the string, compressed and held by the button, and what counts is how it bends in those conditions, what archers call dynamic spine. That the bench measurement is not enough to predict behaviour in the shot is something three models over twenty-five years agree on. In the first, the static test gives critical forces a third lower than those the arrow meets on the string, and the classic rule linking draw weight to the static critical force picks the shaft the dynamic analysis rejects, because the arrow's mass does not enter the static test. In the second, an X10 550, 4% stiffer than a ProTour 570, needs less draw weight to leave cleanly, because the two shafts have different stiffness profiles along their length: shafts of the same nominal spine but of different models should not be taken as equivalent. Nobody has yet shot the same shafts to compare the static ranking with the ranking in the shot, so in the registry the claim is transferred and rests on the models.
The same calculations explain why two archers can take the same shaft off the shelf and need different arrows. That spine has to be matched to draw weight and arrow length holds in every model that has studied it, each reaching it by a different route, and their figures give an idea of how narrow the window is. Four factors are at play.
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Draw weight
The more powerful the bow, the more the arrow flexes, and the same shaft behaves as if it were weaker. In a model of the target recurve, moving to the adjacent spine takes about 11 newtons of draw weight, two and a half pounds, while a bow can only be adjusted within about 10% of its nominal weight: at a given draw length, then, only one or two shafts can be tuned well.
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Arrow length
All else being equal, a longer shaft flexes more: in the same model ten extra millimetres of shaft are worth about four and a half newtons less draw weight. Since a shaft can be cut but not lengthened, when in doubt it makes sense to start a little long and cut after testing.
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Point weight
A heavier point makes the arrow behave as a weaker one, and it is the lever most used on the line for fine tuning. The only model that has calculated it, though, finds that it moves things very little: going from 120 to 100 grains is worth half a newton of draw weight, against three newtons for vanes going from 11 to 3 grains. That weight at the tail matters more remains a result of that model alone, never checked on the bench or on the target.
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Release type
With the fingers the string slides sideways across the fingertips, with a mechanical release hardly at all, and the arrow flexes much less, as shown further on. That is why recurve and compound are tuned differently.
The practical point is that an arrow is only right in relation to the system you shoot it in. Change the poundage, the point or the length and you change how it behaves in the shot, with the same shaft in your hand, and a tweak to the equipment that seemed harmless can move the group without your having touched your technique.
When the spine does not match: reading the arrows
The starting point is the manufacturers' charts, which cross draw weight with arrow length. In the footage of the sixteen elite archers the spine was in every case the one the manufacturer's chart gave for their draw weight and length, but in all likelihood the shafts had been chosen from those very charts: the data say that the best archers follow them, not that the charts point to the best shaft. A 2025 model, not yet tested with shots, reconstructs them overall but finds them too stiff, by up to four sizes, for long arrows and low draw weights, and notes that they move in one-inch steps while in many rows half an inch of arrow already changes the recommended shaft.
From the chart you refine on the range. The classic method is the bare shaft test: from the same distance you shoot a few fletched arrows and a few without vanes, and look at where one group lands relative to the other. Without the fletching to straighten it in flight, the arrow shows the spine error that the vanes would otherwise cover up.
The direction needs reading carefully, because it changes with the hand and the type of release. For a right-handed archer shooting with the fingers, the rule used in tuning is this: if the bare shaft lands to the left of the fletched group the arrow is too stiff, meaning it does not flex enough; if it lands to the right it is too weak. For a left-handed archer the directions are reversed. On the recurve the pressure button, or plunger, also comes into play, and with its spring and its position it adjusts exactly this behaviour. When the fault is large enough you can see it with the naked eye in flight: it is called fishtailing. One further sign comes from the bow after the release: in the shot where the arrow left the string closest to the median plane the string stopped almost at once, and according to the authors of the footage, with the bow set up like this, the archer should notice a quieter bow and a different feel.
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Too stiff
Spine number too low for your bow
It does not flex enough. For a right-handed archer shooting with the fingers, the bare shaft lands to the left of the group.
To weaken it: a heavier point, a slightly longer arrow, or a few more pounds on the bow.
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Too weak
Spine number too high for your bow
It flexes too much. For a right-handed archer shooting with the fingers, the bare shaft lands to the right of the group.
To stiffen it: a lighter point, a slightly shorter arrow, or a few pounds less.
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Matched
Spine that suits the system
The bare shaft lands with the fletched arrows, or very close to them.
This is the condition from which every other adjustment starts.
The corrections all work on behaviour in the shot rather than on the shaft as such, and they are made in small steps: one change at a time, with a test after each. Changing three things at once and hoping the group improves is the surest way to get lost, because you will never know which of the three made the difference. There is one limit to bear in mind, though: the models show a weak shaft slamming into the button or touching the bow with its vanes, but the groups of archers shooting a matched and an unmatched spine have never been compared.
Modern bows: the paradox shrinks, it does not vanish
The paradox comes from bows in which the arrow has to pass beside the grip, and modern bows have reduced it in two ways. The first is the centre-shot riser: the window is cut to the centre or beyond, so the arrow sits very close to the plane of the bow and the angle at which it leaves gets smaller. The second is the mechanical release, which removes the string's slide across the fingers. On the same arrows, with a recurve mounted on a robot, a mechanical release gives a flex of 1.22 millimetres just clear of the bow, two Japanese national-level archers 6.86 and a mechanism imitating the fingers 14.8. That with a release the arrow flexes much less is reported from this single test, with two archers and 72 shots in all, which does not say where the arrows landed.
On the compound the picture points the same way. A model checked against photographs of a single-cam compound mounted on a machine finds a peak flex of a few millimetres before the arrow leaves the string, against the 10 to 15 the same author expects with the recurve, and the arrow enters the air almost aligned, at an angle of a few tenths of a degree. That the arrow flexes less with the compound remains transferred, though, because the comparison with the recurve is not made in the same test and the alignment depends on the tuning, the cams and how much the archer twists the bow; years later the same author writes that design differences between compounds make a general rule hard to give.
Reduced does not mean gone: even with a release the arrow flexes, and spine still matters, only with more margin. On the same compound the recommended arrow, a ProTour 470, and two stiffer sizes, 420 and 380, shot by a machine at 35 metres all landed in small, similar groups, well inside a 40-millimetre circle, with only the sight mark changed. That with a compound and a release, arrows stiffer than recommended group like the recommended one is reported from this single study, with one bow, one shaft model and the machine in place of the archer, and the author infers rather than measures that the recurve is less forgiving. The flex that the release removes is not purely a gain, either: in the Japanese experiment mentioned above the air drag was less repeatable with the release than with the archers.
What you do with this, in practice
When the arrows go off to the side for no obvious reason, before looking for the fault in your technique check that the spine suits your current equipment. You start from the manufacturer's chart or a spine calculator, where you enter draw weight, draw length and point weight, and from there you refine with the bare shaft test and the button, one change at a time. How to put the whole tuning process in order I explain in the article on bow tuning.
It works the other way round too: every time you change something important, the poundage of the bow, the point weight or the arrow length, the way the arrow bends in the shot changes as well. The arrow that suited you six months ago may no longer do so if you have raised the poundage in the meantime, and since only one or two shafts tune well at a given draw length, it pays to check it again after every major change.