Sectional Density - Desert Dog

[Jefferry404]

Yes, SD is a number. But there is an associative practicality to the number. Please see my previous post.
Take a 450 grain and 350 grain .416.
If one reduces the practical velocity of the 350 to the same as the 450, then yes, the SD does not matter per se. if the manufacturer bullet velocity envelopes were designed the same, then the expansion and hence penetration, the same.
But who would do this? And consider the energy is about 29% lower!
Then practicality comes in and the lighter bullet has higher velocity and if the envelopes are the same, i.e. Woodleigh, then the lower SD penetrates less.
QED

 

[DaddyFlip]

@Miletic SD does not participate in the construction of a projectile; it's merely the ratio of weight to diameter regardless of construction.

@Jefferry404 I would argue that there's no practicality to the SD number. The lower SD bullet penetrates less because it weighs less; the SD value is merely the resulting calculation, not the reverse.

As it stands, if you want to increase weight in the same caliber, you must increase length, which reduces case capacity and limits velocity. The SD of a projectile doesn't change with length, only weight and diameter. The other limit on velocity is chamber pressure. We already know that replacing lead with solid copper or brass increases length at the same weight because the materials are less dense.

But what if you went to a more dense material than lead, like tungsten, and kept the same bullet length so that case capacity were constant. Now you've increased the SD at the same caliber, bullet length, and case capacity; all of a sudden, SD has the potential for meaning. If your more dense bullet can hold together, then it theoretically will penetrate more (I'm discounting, for a moment, the well known function of hydrodynamics in the fleshly, mammalian body). However, there is a limit to how much more velocity can be achieved with a heavier bullet, as @Jefferry404 alluded to above while describing the opposite phenomenon. So because there is a maximum operating pressure for any given cartridge, there is a practical limit to the diameter-weight-velocity relationship of a cartridge. Heavier bullets require lower velocity regardless of their design or material of construction. Therefore, we're back to SD having no meaning.

I'll restate my original post in this thread. Sectional density is a calculation of no consequence. No one to my knowledge or memory talked about sectional density until recently. It is being hyped now to market smaller caliber cartridges of equal weight to larger calibers in fast twist modern rifles. That's my observation and opinion.

 

[Jefferry404]

I suggest getting Schwartz's "Quantitative Ammunition Selection". It is not expensive and the gentleman develops the math and physics, not just words. I use it for relative comparison of reloads. Yes, on its own, SD is a number but as pointed out, the caliber, weight and velocity are usually correlated in the sense that one cannot get any velocity for any weight and any diameter, so SD comes in indirectly. The formula has impact velocity, terminal bullet mass, diameter, bullet yield strength, the test medium density and a drag coefficient in the medium.

Anyhow, a chart for those that need pics and the normalized sensitivity in the table for the others

 

[Johnny_2_Knives]

I stopped watching this guy when he filmed his wife going to the bathroom when on a hunt in Africa...

 

[1dirthawker]

I'll restate my original post in this thread. Sectional density is a calculation of no consequence. No one to my knowledge or memory talked about sectional density until recently. It is being hyped now to market smaller caliber cartridges of equal weight to larger calibers in fast twist modern rifles. That's my observation and opinion.

slight disagreement here.

first, sectional density is a calculation of no consequence, i would say with todays better bullet tech that that is more correct than it use to be, however in the early 1900's using cup and core bullets, sectional density was an indicator of bullet performance when using softs on game.

also, sectional density does change the moment the bullet hits flesh, the bullet expands and its diameter immediately changes as does its sectional density, so there are two types of section density, one with the bullet sitting in the cartridge case and the other when it hits flesh and bone and starts expanding (terminal SD). of course with a good solid, sectional density does NOT change but it is generally an INDICATOR of its ability to penetrate.

second, gun writers have discussed sectional density for decades (yeah, I'm 65 now and remember them talking about it, especially it it drove home a point they were in favor of, like deep penetration). in fact the SD of the 458 winchesters 500 gr bullet was touted as a reason for its great penetration. maybe SD is being hyped today as well for the same reasons.

i always looked at SD the same as i did foot pounds of energy. ft lbs is a indicator of the relative power of a cartridge but taken by itself is a worthless indicator of actual performance on game. for instance the 22-250 vs 45-70 both create similar ft lbs of energy but one of those performs vastly better on big game than the other. SD is similar, used by itself to determine effectiveness of a bullet is a flawed idea but it does give a relative indication of the bullets penetration abilities.

as noted above, just my observation and opinion.

 

[Bob Nelson 35Whelen]

Within the same bullet design and construction, sectional density absolutely matters.

Let’s compare .375 caliber monolithic bullets same brand and model:

A 270‑grain mono will generally penetrate deeper than a 250‑grain mono.

In my thought process, why Sectional Density Matters…..

With monolithic bullets

• Weight retention is nearly 100%
• Expansion is controlled and very consistent
• Construction is identical

Because those variables are essentially held constant, the remaining differences are:

1. Sectional density
   • 270 gr (.375) SD ≈ 0.274
   • 250 gr (.375) SD ≈ 0.254
2. Momentum
   • The heavier bullet carries more momentum at similar velocities
3. Resistance vs mass
   • Same expanded diameter
   • More mass behind that frontal area

Result: Higher SD + more momentum = greater penetration

@CBeck
I can agree with you when it comes to monos but it's a different story with cup and core either bonded or non bonded.
I say this because of the following
When a lead core bullet of any nose profile contacts game it begins to expand.
Let's take the typical 30 cal 180gn. These typicality expand 1.5-2 times diameter.
Once these bullets expand they then become 50-60 cal projectiles that for argument sake still weigh 180grains. A 60 cal 180 gn projectile has a very low SD compared to the original 30 cal.
So SD has now changed dramatically and is no longer relevant.
As long as a projectile can penetrate to the vitals and cause catastrophic life threatening organ and tissue damage it will kill an animal. This damage can also occur thru secondary fragmentation as the projectile loses mass. As soon as a projectile contact game SD changes.
Bullet construction, velocity and mass all combine to kill game. SD may give an indication of a projectiles capability but the more it expands the more SD changes to a lower number.
Just my humble opinion.
Bob

 

[Nhoro]

I would never challenge anyone's personal experience and you are almost making my mathematical point. You are speaking of increasing mass within the same diameter, and there is no argument against that. The 'modern' proponents of SD suggest that a smaller diameter of equal mass is better. Here are a handful of potential/traditional 300gr African calibers in descending order of SD; what is your choice for buffalo if bullet construction is identical? The highest SD is 338 Win Mag; the lowest is 458 Win Mag.

338 Win Mag
9.3 x 62
375 H&H
405 Winchester
416 Rigby
458 Win Mag

Quite the opposite.

• .416 Rigby (400-grain bullet): .330 SD
  • Note: 410-grain bullets have an SD of .338
• .458 Winchester Magnum (500-grain bullet): .341 SD
• .338 Winchester Magnum (225-grain bullet): .281 SD
  • Note: 250-grain bullets have an SD of .313
• .375 H&H Magnum (300-grain bullet): .305–.320 SD
• 9.3x62mm (286-grain bullet): .305 SD

 

[dchum]

Quite the opposite.

• .416 Rigby (400-grain bullet): .330 SD
  • Note: 410-grain bullets have an SD of .338
• .458 Winchester Magnum (500-grain bullet): .341 SD
• .338 Winchester Magnum (225-grain bullet): .281 SD
  • Note: 250-grain bullets have an SD of .313
• .375 H&H Magnum (300-grain bullet): .305–.320 SD
• 9.3x62mm (286-grain bullet): .305 SD

He said all with 300 gr bullets.

 

[DaddyFlip]

slight disagreement here.

first, sectional density is a calculation of no consequence, i would say with todays better bullet tech that that is more correct than it use to be, however in the early 1900's using cup and core bullets, sectional density was an indicator of bullet performance when using softs on game.

also, sectional density does change the moment the bullet hits flesh, the bullet expands and its diameter immediately changes as does its sectional density, so there are two types of section density, one with the bullet sitting in the cartridge case and the other when it hits flesh and bone and starts expanding (terminal SD). of course with a good solid, sectional density does NOT change but it is generally an INDICATOR of its ability to penetrate.

second, gun writers have discussed sectional density for decades (yeah, I'm 65 now and remember them talking about it, especially it it drove home a point they were in favor of, like deep penetration). in fact the SD of the 458 winchesters 500 gr bullet was touted as a reason for its great penetration. maybe SD is being hyped today as well for the same reasons.

i always looked at SD the same as i did foot pounds of energy. ft lbs is a indicator of the relative power of a cartridge but taken by itself is a worthless indicator of actual performance on game. for instance the 22-250 vs 45-70 both create similar ft lbs of energy but one of those performs vastly better on big game than the other. SD is similar, used by itself to determine effectiveness of a bullet is a flawed idea but it does give a relative indication of the bullets penetration abilities.

as noted above, just my observation and opinion.

I appreciate your response and hitting on several related factors. To continue the conversation, I'm going to rebut and agree where I can, in no particular order:

• "with today's better bullet tech"; "cup and core bullets" - SD doesn't care about the shape, style, bonding, material, length, base, or any other feature of a bullet. Only weight and diameter matter. The same applies today as it did in the 1900's
• The reason the 458WM 500gr bullet had great penetration, all other things equal, was because it was the heaviest 458WM bullet; SD may have been touted by a gunwriter or marketing person, but it was not the reason. A 480gr, 450gr, 400gr, 300gr bullet would penetrate less because it weighs less; the SD calculation was just along for the ride. You don't even need it as an indicator if you understand that heavier is better- like our forefathers did before these calculations came along. If you're a gunwriter trying to recruit followers to yourself or your magazine or a bullet or cartridge manufacturer trying to differentiate your product, you might tout SD.
• Unfortunately, once a bullet penetrates a target, the SD cannot be known so it is irrelevant in the SD conversation we're having. Within a caliber (let's stay with 458 since we're already talking about it), the SD always increases in direct proportion to bullet weight (because the other factor, diameter, is constant), so we can easily say that the true indicator of penetration potential is bullet weight. We don't need SD to tell us that; therefore, it is inconsequential.
• Only a decade separates us, and your memory/attention may be better than mine, but I don't recall SD ever being as big a subject as, say, BC or bullet construction was in the 80's-2000.
• What happens if we talk about a single bullet weight across several calibers? I showed earlier where a 250gr 458 bullet (for 45-70) has an SD of 0.170 while a 250gr 338 bullet has an SD of 0.313 (as diameter gets smaller, SD increases in proportion). In this case, SD does give an indication of relative penetration potential because there's equal weight behind a smaller diameter bullet- the weight is concentrated on a smaller contact area so the pressure (psi) is greater.
• Bumping up to 300gr (to eliminate the 45-70 and add the 416) here are the SD's for the following cartridges. For practical reasons, the big game hunter probably would not choose the 338 or 458 in 300gr. What about the middle three? We already know from the previous point that smaller diameter at the same weight concentrates more of that weight on a smaller area, increasing psi and increasing penetration potential; however, the selection of rifle, caliber, bullet style, and all those other things @michael458 had on his list far outweigh SD in the process (it's at the bottom of the list- it's just a calculation).
  • 0.375 (338 WM)
  • 0.320 (9.3x62/74R)
  • 0.305 (375 H&H)
  • 0.248 (416 Rem)
  • 0.204 (458 WM)
• What happens if we keep the SD constant and see what calibers and bullet weights are close to it? Now the relationship is similar to muzzle energy; we can find some 223 bullets that have the same SD as some 416 and 458 bullets - that doesn't tell us anything practical or useful. This, along with the above illustrations, is why I believe SD is of no consequence. BC is of consequence as it involves so many other factors the hunter/shooter can't account for on his own. SD? Irrelevant. Just use bullet weight.

 

[Miletic]

I hope you won't be angry if I interfere.
Every body that moves with the room carries a certain energy. Energy is expressed in Joules (J). It is calculated as m x v2 / 2.
From this formula we see that the most important factors for energy are mass and speed. The movement of projectiles fired from firearms is defined by a branch of physics called ballistics.
It is divided into internal (follows the movement of the projectile inside the barrel), external (movement from point A to point B outside the barrel) and terminal (movement through the target (certain medium)). As this is a topic related to the penetration of projectiles through a certain medium, we are interested in the terminal ball. How far the projectile will penetrate the obstacle depends on its shape, its composition, its mass and speed (not the initial speed at the mouth of the barrel, but the speed during contact with the obstacle).
The narrowest projectile with the highest mass and the highest speed will achieve the deepest penetration. There are two types of projectiles from firearms: 1. humane (full muzzle) 2. expansive (soft tip of the projectile). We are interested in exp.
When it comes to explosive bullets, a new phenomenon called deformation of the designed shape appears as an obstacle in penetration. This deformation can be random and controlled.
Most often, when a projectile hits an obstacle at high speed, its diameter increases, which is a consequence of the other two disciplines in physics, which are called action and reaction. The reaction force is equal to the action force (Newton's third law).
In order to delay the reaction force (to make it happen as soon as possible) it is necessary for the projectile to keep its original shape as long as possible. And that's where we come to SD.
SD is important and pronounced with soft point projectiles that do not have controlled expansion, but the manufacturer wants them to make a partial increase in diameter when hitting an animal, without disintegrating.
Disintegration or loss of mass during penetration significantly affects the loss of speed and thus the depth of penetration. The loss of mass is largely solved by the fact that the blade is connected to the lead core (bonding or mechanics). But, there is still a loss of mass from the center of the lead core, because a projectile usually loses mass from its center, from its center, which is a chemical element. It is here that the importance of element density, which is expressed as SD, comes into play. In the case of "goose" projectiles, which are actually of better quality, the penetration will be higher.
This also applies to some extent to monolithic grains, but it is far less pronounced. The basis is speed and mass retention followed by diameter expansion. One more important thing. SD can have something to do with the weight of the projectile (for example, two projectiles of the same lead and the same size, and one is heavier), but it doesn't have to. The weight of a lead projectile can depend on the amount of antimony and tombac that is inserted into it. SD code for projectiles fired from a firearm should be accepted conditionally as the quality of the material, so it affects the penetration in that way.

 

[CBeck]

@CBeck
I can agree with you when it comes to monos but it's a different story with cup and core either bonded or non bonded.
I say this because of the following
When a lead core bullet of any nose profile contacts game it begins to expand.
Let's take the typical 30 cal 180gn. These typicality expand 1.5-2 times diameter.
Once these bullets expand they then become 50-60 cal projectiles that for argument sake still weigh 180grains. A 60 cal 180 gn projectile has a very low SD compared to the original 30 cal.
So SD has now changed dramatically and is no longer relevant.
As long as a projectile can penetrate to the vitals and cause catastrophic life threatening organ and tissue damage it will kill an animal. This damage can also occur thru secondary fragmentation as the projectile loses mass. As soon as a projectile contact game SD changes.
Bullet construction, velocity and mass all combine to kill game. SD may give an indication of a projectiles capability but the more it expands the more SD changes to a lower number.
Just my humble opinion.
Bob

No argument from me on any of your thoughts …

Which is to say that’s why I phrased it as…..”Within the same bullet design and construction”

 

[DaddyFlip]

You provided a lot of good background information on ballistics. Let me ask you to clarify a couple of points for my understanding:

SD is important and pronounced with soft point projectiles that do not have controlled expansion, but the manufacturer wants them to make a partial increase in diameter when hitting an animal, without disintegrating.

I think what you are saying here is that for projectiles that are not expanding, the calculated SD can be relied on more because the projectile maintains most, if not all, of it's original diameter? As opposed to an expanding projectile where the SD changes so rapidly that the calculated SD is not important?

It is here that the importance of element density, which is expressed as SD, comes into play. In the case of "goose" projectiles, which are actually of better quality, the penetration will be higher.

SD can have something to do with the weight of the projectile (for example, two projectiles of the same lead and the same size, and one is heavier), but it doesn't have to. The weight of a lead projectile can depend on the amount of antimony and tombac that is inserted into it. SD code for projectiles fired from a firearm should be accepted conditionally as the quality of the material, so it affects the penetration in that way.

What do you mean by "goose"?

SD always has something to do with the weight of the projectile; the calculation is the weight divided by the base area. Yes, you can have two bullets of the same diameter, shape and size with one heavier because the material has a greater density, but this will, by definition, increase the weight, which by calculation will increase the SD.

Based on many of the responses, it appears some believe that SD is the root factor and weight is the consequence; the opposite is true. SD is the consequence of weight and diameter. It's a ratio of weight to area; as the number gets higher, it says that there's more weight behind the same area.

 

[Miletic]

You provided a lot of good background information on ballistics. Let me ask you to clarify a couple of points for my understanding:

I think what you are saying here is that for projectiles that are not expanding, the calculated SD can be relied on more because the projectile maintains most, if not all, of it's original diameter? As opposed to an expanding projectile where the SD changes so rapidly that the calculated SD is not important?


What do you mean by "goose"?

SD always has something to do with the weight of the projectile; the calculation is the weight divided by the base area. Yes, you can have two bullets of the same diameter, shape and size with one heavier because the material has a greater density, but this will, by definition, increase the weight, which by calculation will increase the SD.

Based on many of the responses, it appears some believe that SD is the root factor and weight is the consequence; the opposite is true. SD is the consequence of weight and diameter. It's a ratio of weight to area; as the number gets higher, it says that there's more weight behind the same area.

Thank you for reading all that I wrote
1. Yes,I think that with monoliths and projectiles that do not deform, you can rely on SD
2. goose is density, I apologize, I stated wrongly, Eng is not my basic language
3. I agree with you on how SD is calculated and that it is inevitably connected to the mass and the space in which that mass is located
4. I also agree with you that SD is a consequence of the weight and space in which that mass is located

 

[DaddyFlip]

First, I want to thank the forum for allowing me to dominate the latter part of this thread with my attack on sectional density (SD) as an important hunting bullet consideration. This is a great forum of folks and I appreciate how we are able to keep (most) discussions civil, educational, and entertaining.

Second, I want to address @Beck's OP questioning DD's opinion of SD. To this point, I had only watched the first few minutes of the video where he initially dismissed SD because I agreed 100% with that assessment. All my posts have been based on my own knowledge and understanding of math, ballistics, bullet design, and marketing. I just watched the complete video and can say that DD is 100% correct in everything he said. DD said one purpose of the video was to, "break you from the shackles of SD"; I agree that shooters and hunters should dismiss SD completely. He said something else that I will need to research for myself, but I believe him: "Sectional density was developed to optimize military artillery, not to describe a hunting bullet."

For fun, I put together a spreadsheet for your consideration. If you had to guess what the average SD is for the weight and primary caliber of every PG/DG hunting diameter (0.257 - 0.577), what would you guess? What if I told you it was 0.305? Would it be a surprise if I told you that the SD of the 375 H&H 300gr bullet was 0.305? There must be something special about mediocrity... I mean about being in the middle. Consider the following table:

The first column is obviously bullet diameter. I've presented a representative caliber because I was going to calculate loads that might equalize the energy between the calibers, but I got tired. The third column is important; it takes the SD of 0.305 for the 300gr 375 H&H and outputs the exact bullet weight every other caliber would need to be to give the same SD of 0.305. For better or worse, we live in an imperfect world, so we don't have bullets available to us in those exact weights. The next column shows what the typical or closest bullet weight is for each caliber and then the actual SD is calculated for that weight in the next column. Finally, the deviation from 0.305 is shown in the final column.

Everyone is free to draw their own conclusions; I merely present the data for entertainment purposes. I will point out a couple of comparisons for fun. The 7x57 Mauser may be capable of better penetration than the 375 H&H. Is the hunter going to choose the Mauser for buffalo or is he going to choose it as an alternative to the 375 for sable or impala? On the opposite scale, should the 505 Gibbs owner leave his rifle at home and instead rent the PH's 375 for elephant?

As another aside, I was reading through Hornady's 11th edition in preparation for my earlier posts in this thread. I just looked through again at their detailed bullet descriptions (they are a bullet manufacturer and retailer) and not once did they mention sectional density (SD). They mentioned ballistic coefficient (BC) many times, but not SD. Sectional density does not matter; it's not made up, but for the hunter it IS made up. It is of no consequence. It is easy to select the right bullet for the task at hand without ever thinking about sectional density. Heavier animals require heavier bullets that hold together on impact. Lighter animals may require bullets that fragment on impact. Sectional density is not going to advise you on the correct bullet to select for the animal you wish to harvest.

 

[1dirthawker]

BC is of consequence as it involves so many other factors the hunter/shooter can't account for on his own. SD? Irrelevant. Just use bullet weight.

well, the point is.... sectional density is entirely related to bullet weight ( assuming the same caliber)

 

[DaddyFlip]

well, the point is.... sectional density is entirely related to bullet weight ( assuming the same caliber)

Yes it is... and (assuming the same caliber), sectional density is entirely irrelevant.

 

[1dirthawker]

Yes it is... and (assuming the same caliber), sectional density is entirely irrelevant.

well,

yes and no, depending on bullet construction as i tried to relay earlier.

for example take a SD of .173 and .342 in the same caliber with a cup and core OR a mono metal bullet (we are just looking at same bullet construction here) is SD irrelevant?

 

[DaddyFlip]

SD doesn't care about bullet construction.

I'm looking for a caliber that has a real bullet SD of 0.173 and 0.342; I can't find one. The closest I can find with that spread is 458 with a 250gr at 0.171 and a 500gr at 0.341. Within the SAME caliber and construction, SD is irrelevant; the only thing that matters is bullet weight because it is known that heavier bullets of the same construction and diameter will penetrate deeper. Also doesn't matter if it's C&C or MM; if it's the same diameter and weight, the SD is the same. Any other bullet performance characteristic (expansion, for example) is entirely detached from the SD calculation.

 

[Bob Nelson 35Whelen]

First, I want to thank the forum for allowing me to dominate the latter part of this thread with my attack on sectional density (SD) as an important hunting bullet consideration. This is a great forum of folks and I appreciate how we are able to keep (most) discussions civil, educational, and entertaining.

Second, I want to address @Beck's OP questioning DD's opinion of SD. To this point, I had only watched the first few minutes of the video where he initially dismissed SD because I agreed 100% with that assessment. All my posts have been based on my own knowledge and understanding of math, ballistics, bullet design, and marketing. I just watched the complete video and can say that DD is 100% correct in everything he said. DD said one purpose of the video was to, "break you from the shackles of SD"; I agree that shooters and hunters should dismiss SD completely. He said something else that I will need to research for myself, but I believe him: "Sectional density was developed to optimize military artillery, not to describe a hunting bullet."

For fun, I put together a spreadsheet for your consideration. If you had to guess what the average SD is for the weight and primary caliber of every PG/DG hunting diameter (0.257 - 0.577), what would you guess? What if I told you it was 0.305? Would it be a surprise if I told you that the SD of the 375 H&H 300gr bullet was 0.305? There must be something special about mediocrity... I mean about being in the middle. Consider the following table:
View attachment 748864

The first column is obviously bullet diameter. I've presented a representative caliber because I was going to calculate loads that might equalize the energy between the calibers, but I got tired. The third column is important; it takes the SD of 0.305 for the 300gr 375 H&H and outputs the exact bullet weight every other caliber would need to be to give the same SD of 0.305. For better or worse, we live in an imperfect world, so we don't have bullets available to us in those exact weights. The next column shows what the typical or closest bullet weight is for each caliber and then the actual SD is calculated for that weight in the next column. Finally, the deviation from 0.305 is shown in the final column.

Everyone is free to draw their own conclusions; I merely present the data for entertainment purposes. I will point out a couple of comparisons for fun. The 7x57 Mauser may be capable of better penetration than the 375 H&H. Is the hunter going to choose the Mauser for buffalo or is he going to choose it as an alternative to the 375 for sable or impala? On the opposite scale, should the 505 Gibbs owner leave his rifle at home and instead rent the PH's 375 for elephant?

As another aside, I was reading through Hornady's 11th edition in preparation for my earlier posts in this thread. I just looked through again at their detailed bullet descriptions (they are a bullet manufacturer and retailer) and not once did they mention sectional density (SD). They mentioned ballistic coefficient (BC) many times, but not SD. Sectional density does not matter; it's not made up, but for the hunter it IS made up. It is of no consequence. It is easy to select the right bullet for the task at hand without ever thinking about sectional density. Heavier animals require heavier bullets that hold together on impact. Lighter animals may require bullets that fragment on impact. Sectional density is not going to advise you on the correct bullet to select for the animal you wish to harvest.

@DaddyFlip
A 275 is readily available for the 35 Whelen in Australia so that brings it pretty close to the magical zero of the 375. The sd if that bullet is .307
Just saying.
Bob

 

[DaddyFlip]

@DaddyFlip
A 275 is readily available for the 35 Whelen in Australia so that brings it pretty close to the magical zero of the 375. The sd if that bullet is .307
Just saying.
Bob

I remember seeing a 270ish bullet for the Whelen somewhere but I couldn't confirm it yesterday when I was making the chart.