XVIII. SIMPSON'S RULE.

193. We shall now give a very important Rule by which the areas of certain figures may be approximately

found.

Let there be an area bounded by the straight line AG, the straight lines Aa and Gg at right angles to AG, and the curve ag.

Divide AG into any even number of equal parts AB, BC, CD, .; at the points of division draw straight lines Bb, Cc, Dd, ... at right angles to AG, to meet the curve. The straight lines Aa, Bb, Cc,

Gg are called ordinates.

FG

Then the area may be approximately found by the following RULE: Add together the first ordinate, the last ordinate, twice the sum of all the other odd ordinates, and four times the sum of all the even ordinates; multiply the result by one-third of the common distance between two adjacent ordinates.

194. In the figure there are seven ordinates; the even ordinates are Bb, Dd, and Ff; the other odd ordinates besides the first and last are Cc and Ee.

195. The Rule given in Art. 193 is sometimes called the method of equidistant ordinates; but it is more usually called Simpson's Rule, although it was not invented by Simpson.

196. Examples:

(1) Suppose there are seven ordinates, the common distance being 1 foot; and that these ordinates are respectively 4:12, 4:24, 4·36, 4·47, 4·58, 4'69, and 4.80 feet.

Thus the area is about 26'8 square feet.

(2) Suppose there are five ordinates, the common distance being 2 feet; and that these ordinates are respectively 126, 144, 1·59, 1·71, and 1·82 feet,

197. In the figure of Art. 193, the curve is concave towards the straight line AG; moreover the ordinates continually increase from one end of this straight line to the other. But Simpson's Rule is applicable to areas in which the curve has other shapes, as in the following figures:

The result will in general be more accurate the more ordinates are used; and the Rule ought not to be trusted if the curve be very irregular.

198. If the area be bounded by the straight line AG and the curve AdG, the same Rule applies; only here the first and last ordinates are no- A thing, and so do not occur in the calculation.

If the area be bounded by

BCD E F

β Y 8 e

the closed curve AdG8A, the same Rule may be applied: we have now to take as the ordinates the breadths bß, cy, dd, ...

199. The beginner would not be able to follow a strict investigation of Simpson's Rule;

but he may see without difficulty that there are grounds for confidence in the approximate accuracy of the Rule.

We will call the portion of the area between two consecutive ordinates a piece; and we will consider the first two pieces of the figure in Art. 193.

A B

Suppose the straight line ab drawn; we thus obtain a trapezoid, and it is obvious from the diagram that the first piece is greater than this trapezoid; so that the area of the first piece is greater than the product of AB into half the sum of Aa and Bb.

Therefore twice the area of the first piece is greater than the product of AB into the sum of Aa and Bb.

Similarly, twice the area of the second piece is greater than the product of BC into the sum of Bb and Cc.

Thus twice the area of the first two pieces is greater than the product of AB into the sum of Aa and Cc and twice Bb.

Again: suppose the straight line bt to touch the curve at b, and let Aa and Bb be produced to meet this straight line. Thus another trapezoid is formed; and by Art. 163,

the area of this trapezoid is equal to the product of AC into Bb, that is, to the product of AB into twice Bb. And it is obvious from the diagram that the sum of the first two pieces is less than this trapezoid.

Thus the area of the first two pieces is less than the product of AB into twice Bb.

Hence we may expect that when we combine these two results, the errors will to some extent balance each other; and that three times the area of the first two pieces will be very nearly equal to the product of AB into the sum of Aa and Cc and four times Bb.

By proceeding in this way with the other pairs of pieces in the figure of Art. 193, we may obtain a sufficient confidence in Simpson's Rule. With respect to the figures in Art. 197, the process would be similar, though not absolutely the same; the main fact is that we combine two results, one of which is too large, and the other too small, and trust that the errors will to some extent balance each other.

200. If ag were a straight line, instead of a curve, in the figure of Art. 193, Simpson's Rule would give the exact value of the area; but in such a case the whole figure would form a trapezoid, and the area is most easily found by taking the product of AG into half the sum of Aa and Gg. Also if ag is a curve of a certain form the Rule will give an exact result; but we cannot explain in an elementary manner what this form is.

201. It may happen that the boundary of a figure is a curve which is too irregular to allow of the immediate application of Simpson's Rule; in such a case we may adopt the following method: Draw a rectilineal polygon, differing as little as possible from the figure, and determine the area of this polygon exactly; then by Simpson's Rule calculate separately the areas of the portions which lie between the polygon and the figure; add these to the area of the polygon or subtract them from the area of the polygon, according as they fall without or within the polygon, and the final result will be approximately the area of the figure.

202. In Land Surveying it is often necessary to determine the area of a figure which has its boundary composed

A

B

in an irregular manner of curves, and numerous short straight lines; it is found in practice that the area can be obtained with ease and sufficient accuracy by a method of adjustment of the boundary to a more commodious form. Thus, suppose a field to be represented on a plan by the figure ABCD. Draw a straight line from A to B; the small portions lost and gained will obviously balance each other, exactly or very nearly, and the area will thus remain almost or quite unchanged. Similarly, straight lines may be D drawn from B to C, from

C to D, and from D to A, with a like balance of loss and gain. Thus we have a four-sided rectilineal figure equivalent to the original figure; and the area can be easily obtained.

The skill and judgment of the surveyor will be exercised in drawing the straight lines, so that the greatest possible accuracy may be secured. A piece of transparent horn with straight edges is very useful in drawing the straight lines; the horn is placed over the irregular boundary and is shifted about until there appears to be an equal portion on each side of the edge between the edge and the boundary.

203. An experimental method of determining the area enclosed by an irregular boundary may be noticed.

Suppose a field to be represented on a plan by the figure ABCD of Art. 202. Cut the figure out of stout paper or cardboard of uniform thickness, and weigh it in a very delicate balance. Also cut out a square inch from the same paper or cardboard, and weigh it. Then by proportion we can find how many square inches the figure ABCD contains. And from observing the scale on which the plan is drawn, we shall know the area on the ground which corresponds to a square inch on the plan; and thus finally we can determine the area of the field.

An interesting application has been made of this process to determine the proportion of the water to the land on the surface of the earth. Camb. Phil. Trans. Vol. vi.