To determine if more material is required, we must look at the geometric difference between a straight line (the chord) and an arc (the curve) connecting the same two points.

In your first scenario, the bridge is a straight line of 20 miles. In geometry, the shortest distance between two points is a straight line.

In your second scenario, the bridge follows a curve that rises and falls 266.75 feet over that same 20-mile span. By definition, any path between two points that deviates from a straight line is longer than the straight-line distance. 

To calculate the difference in length ($L_{arc} - L_{chord}$), we can approximate the curve as a parabolic arc, which is standard for gentle swells in civil engineering. For a bridge of length $d = 20$ miles (105,600 feet) and a height (sagitta) of $h = 266.75$ feet, the arc length $s$ is approximately:

$$s \approx d + \frac{8h^2}{3d}$$

Plugging in the values:

$$s \approx 105,600 + \frac{8(266.75)^2}{3(105,600)}$$
$$s \approx 105,600 + \frac{569,322}{316,800}$$
$$s \approx 105,600 + 1.797 \text{ feet}$$

The curved bridge would be approximately 1.8 feet longer than the flat, straight-line bridge. 

Because the bridge is longer, it would require more raw materials—such as steel for the girders, concrete for the deck, and potentially additional reinforcement—to cover the increased distance. Furthermore, the structural engineering of a curved bridge often necessitates different support requirements to account for the geometry of the load, which would likely increase the material usage beyond just the linear increase in deck length. 

Therefore, yes, building over a 266.75-foot swell requires more raw material than a straight, flat span of the same ground distance.