torque multiplication

Lets say I have a car with 42 hp and proportional torque that through gearing takes off acceptably. Lets further postulate that first gear on said engine is @3:1 and the red line for the engine is around 6k rpm.

Is it safe to assume that a large v8 capable of 400hp with appropriate torque and a similar red line would take off nearly as briskly with the transmission in the 1:1 gear as the smaller engine did in first? Neglect the difference in engine weight and assume enough clutch or torque converter slippage to get the v8 to an rpm that has more than 3x the torque of the smaller engine at take off.

I think the only thing you're missing is that your little motor at 3:1 runs though it's rpm range quickly, let's say it gets you to 20 mph in first, while your V8 with the 1:1 will go from 0-60 in the same gear. All else being equal, that's going to be very different as the little motor spins up (moves you up the HP curve faster) while the big motor lugs at a lower RPM and lower in the HP curve for longer. I think you should do this experiment. 

Suppose that we have a CVT that always keeps the engine at the RPM at which it develops peak power.  Since power is just the product of force and velocity, we can figure out the maximum speed at which we can spin the wheels.  If we want to just take off hard, we accelerate slightly less than the force required to spin the tires, so let's assume the maximum speed we can spin the tires is when we stop "taking off briskly".

Also suppose that we have all wheel drive, and our tires have a coefficient of friction of 1.0 (not uncommon).  This means that >1g of acceleration will spin the tires.  If the car weighs 3000lb, this is a force of 3000lbf.

With 40hp:

40hp / 3000lbf = 5mph

With 400hp:

400hp / 3000lbf = 50mph

(Wolfram Alpha is a great tool for typing in mixed-unit math like this)

So, what's the conclusion? Mazdeuce is right, the tiny engine will get you great acceleration up until a very low speed, while the gigantic V8 will keep that acceleration up for much longer.  My math here is a gross oversimplification, but the generalization is mostly correct.  More horsepower doesn't mean you can accelerate harder, it just means you can accelerate harder at higher speed.  Diesel trucks with stump-pulling torque accelerate hard at the bottom but run out of steam quickly because they have lots of torque, but not much power.  Higher power engines keep you pinned to the seat at higher speeds.

What torque does the big v8 produce at 600 rpm?   Is it three times what the little engine produces?

So instead of talking about hp we talk strictly about torque. Assuming the torque of the bigger engine at any point along the torque curve after the clutch is released is more than the peak torque of the smaller engine multiplied by the ratio of the lowest gear than will it accelerate the car at least as briskly?

Put it another way: Will a car with 400 foot pounds of torque and a 3:1 rear accelerate as quickly as one with 300 foot pounds and 4:1 gears if the cars weigh the same and they are both in the same gear? Actually, looking at it that way I think something probably needs to be squared and the function isn't linear.

Maybe 40 foot pounds and 3:1 first more closely corresponds to 40x3^2=360 foot pounds?

The difficulty is that you can't really separate torque from horsepower.  We talk about one and ignore the other, when in reality they're the same thing.

You don't need to square the ratio.  Let's assume we're in a 1:1 gear (usually 4th) to make this easier.  If the engine makes 400 lb-ft, and you have a 3:1 rear end, the torque at the axle will be 1200lb-ft.  With 300lb-ft and 4:1, it will again make 1200 lb-ft at the axle.  So yes, in this case, the two setups will accelerate "the same" at that particular operating point.

So long as the engines make the same power (not torque) at some wheel speed, the acceleration will be the same.  30% more speed and 30% less torque gives the same amount of power, so the acceleration is the same.

1 hp @ 1 rpm = a Berking load of torque!  

Area under the curve. Calculus. It's important. 

I think you should look into how the Koenigsegg Regera works. It's vaguely relevant and interesting regardless. 

Another point to not forget is that the thrust required to accelerate a given weight at a given rate *does* increase exponentially as speed increases. I don't remember what the exponent is, but remember that there is one. So, if the thrust available and the weight are constant, then acceleration will approach zero, even without external factors such as air and rolling resistance.  

(I'm using the term "thrust" because that avoids the horsepower vs. torque argument, and talks about what really matters; force pushing forward)