The Art of Wing Shooting: A Practical Treatise on the Use of the Shotgun ..., William Bruce Leffingwell, 1895
http://books.google.com/books?id=e34EmE3tkfkC
The king of all pigeon shooters is Capt. John L. Brewer of Philadelphia.
A gun for pigeon shooting should weigh from 7 1/2 to anywhere under 8 pounds. It should be heavy at the breech, with a long and very straight stock, having a drop from 2 to 2 1/2 inches at the butt. These straight stocks are desirable, because they cause one to shoot high, something which is essential, as there is a tendency to undershoot, and nearly every pigeon is missed by shooting under or behind. A pigeon gun should be bored to shoot a trifle high. Birds going from the trap are almost invariably rising, and as there is an almost uncontrollable tendency to aim at a straight-away bird, one should hold a trifle over.

"Straight" here seems to apply to drop; can't tell from this studio image the grip he used but the position of his hand looks like a pistol grip.