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And Douglas doesn't load his .45/70s gently, my knuckles still smart. If you were to hunt big game in all corners of the world with a single rifle, one chambered for a big case .375 will be appropriate where ever big game is hunted. While interest in it seems to have waned, anything said about a .375 also applies to the .416. The .416s do everything the .375s do, but they ensure you'll spend more money, and absorb more recoil doing them. I like .458s too, but the only way to make a .458 as versatile as a .375 is to mate it to a big Rigby or Weatherby case, and I don't see folks lining up to shoot a .460 Weatherby prone, although now and again you see one for sale with a box of 19 unfired cartridges, so maybe it was attempted.
Now the .45/70 has its place. If you hunt in tough terrain, where the advantages of carrying a slim lever action rifle can be appreciated, and where you might need the short range, knock em flat performance of a heavy for caliber hard cast WFN, to put meat in the freezer or get you out of a fix, then an '86 Winchester or a '95 Marlin is for you. But those circumstances do not fit the definition of typical big game hunting. So in the context of the modern hunting experience, the .45/70 is a niche cartridge.
Its prudent to sight rifles with a muzzle velocity of 2200-2300 fps for 150 yards, rifles with a muzzle velocity of 2500-2700 for 200 yards, and rifles with a muzzle velocity of 3000 fps can be zeroed at 300 yards. By comparison, the zero range of the .45/70 is similar as that of a 12 ga slug. There are circumstances where a 12 ga slug is a good choice, but neither is a 12 ga slug a particularly versatile big game cartridge. Lets assume you intend to knock down a pronghorn at a quarter mile. Now maybe the fellow with the .257 Weatherby will be kind enough to point out that a .375 isn't the best tool for the job, but it can spit out a 230 gr bullet at over 3000 fps, so hitting the speed goat is only a marksmanship problem, but the problem is unsolvable with a 12 ga, and nearly so with a .45/70. Today's wilderness is a different place. In many areas, power lines, oil and gas exploration, mining activities, and tree harvesting have opened land and increased the distances at which game can be spotted, and shot from. To say nothing of the natural openings of large marshes, lakes, rock outcrops, and high elevations which provide a commanding view over the country. As a result, cartridges that at their best are short range performers, are less prudent than those which provide the hunter with flat trajectories and ample killing power a quarter mile down range.
The .375 has been lucky for me; the best practical shooting that I've done has been with a .375. It could be that experience influences my opinion, but since I've been able to do things with a .375 that I couldn't do as easily, or at all, with other cartridges, it supports my argument, rather than negates it. To the man noted by bigborefan who does all of his game shooting with a .45/70 Sharps, my hat is off to you. Your level of dedication to the sport, and to the degree of difficulty it entails, few others would consider. That is commendable, I appreciate the commitment; it goes to show that there is more than one correct answer to the question. The next question then is how much time, energy, and money are you willing to invest? Becoming a deadly long range shot with a .45/70 is not an inexpensive prospect, and is not for the new breed who demand instant gratification.