This last case corresponds to the piece of vertical cliff under point J in the section fig. 1.

43. Integration. The general case corresponds to the gradual stepping along the other parts of this section. The length of each step is projected horizontally (87 or 8x) and vertically (sh or dX). The latter is the rise in level, and it equals the gradient multiplied by the horizontal projection of the length of step.

In stepping continuously from one particular point on the section to another, for instance, from A to C on fig. 1, the total horizontal distance between the two is the sum of the horizontal projections (the dl's or dx's) of all the separate steps; and the total difference of level is the sum of the vertical projections (the Sh's, or SL's, or SX's) of all the separate steps. In climbing the hill, the climber rises the whole difference of level from A to C, step by step: the total ascent is the sum of all the small ascents made in all the long series of steps. If the distance be considerable, the number of steps cannot be counted, except by some counting instrument, such as a pedometer; but the total ascent remains the same, whether it be accomplished in an enormous number of extremely short steps or in an only moderately large number of long strides.

The mathematical process of calculating these sums is called Integration.

This mathematical process is indicated by the symbol the Greek capital, when the individual steps are of definitely measurable small size. But when the method of summation employed is such that it assumes the steps to be minutely and immeasurably small, the number of them being proportionately immeasurably large, and when, therefore, of necessity the method takes no account of, and is wholly independent of, the particular minute size given to the steps, then the symbol employed is, which may be looked upon as a specialised form of the English capital S, the first letter of the word "sum." The result of the summation is called the Integral.

44. Increment Symbols. The separate small portions, whose sum equals the Integral, are called the Increments or the Differentials.

When the increments are of definitely measurable small size, they are indicated by the symbols Sx, SX, 8h, SL, SY, etc., etc.

When they are immeasurably minute, and their number correspondingly immeasurably large, they are indicated by the symbols dx, dX, dh, dL, dY, etc., etc.

45. Integration Symbols. Limits of Integration.—The integration is carried out between particular limits, such as B and C in

fig. 1. These limits are sometimes written in connection with the symbols of integration, thus:

If lche be the co-ordinates of the point C, fig. 1, and lÅ be those of the point B, then these integrals mean the same thing as

(heh) or (lc - IB).

The limits are above indicated in the symbol by the names only of the points referred to. The points themselves are, however, frequently indicated only by the values of their co-ordinates, and then it is customary to indicate the limits of integration by writing at top and bottom of the sign of integration the limit-values of the variable whose increment appears in the intregral. Thus, since

dh Sh= 81 dl

we have the integral of Sh between B and C expressible in the two following forms:

If particular points be indicated by numbers, the symbolism becomes somewhat neater. Thus the integral of 8X between the points 1 and 2 of the x, X curve at which points the ordinates may be called X1 X21 and the co-ordinates X, X2, is

Or, again, if it were convenient to call the two limiting values of x by the letter-names a and b, then the same would appear as

Or, if the limiting values of x were, say, 15 and 85 feet, it would

It must be noted that the limits which are written in always refer to values of the variable whose increment or differential appears in the integration. Thus the a and b or the 85 and 15 above mean invariably values of x, not values of X nor of X'. 46. Linear Graphic Diagrams of Integration. In figs. 5 and 6 are given two methods of graphically representing this process of integration. The first corresponds with the illustrations we have already employed. Here the curve xX is supposed to be built up step by step by drawing in each small stretch of horizontal length Sx at a gradient equal to the known mean gradient X' for that length. The gradient X' is supposed known for each value of x, and its mean value throughout each very small length dx is therefore known.

X

X 2

FIG. 5.

With regard to this statement it should be noted that a curve does not really possess a gradient at a point, but only throughout a short length. When we speak of the slope of a curve at a point, what we really mean is the slope of a minute portion

of its length lying partly in front and partly beyond the point: that is, there is actually no difference of meaning between the phrases "the slope of the curve at the point" and "the mean gradient throughout a short length at this point." Since each increment of X, or 8X, equals X' times the corresponding increment of x or Se, we have in fig. 5 all these increments of X

each equal to X'dr projected by horizontal lines on the final vertical X2, where it is perhaps a little more plainly seen that they all sum up to X-X, than without this projection, as in our previous example of climbing a hill step by step as in fig. 1.

2

47. The Increment deduced from the Average Gradient.— If this graphic method be actually employed for practical integration, then care must be taken to let the short stretch drawn at each slope X' lie partly before and partly beyond the vertical drawn at the corresponding distance x. This is necessary, because in actual graphic construction on paper we cannot make the increments & immeasurably small. It is best to let each short curve length lied behind and 8x beyond x.

48. Area Graphic Diagram of Integration.-In fig. 5 the "x-gradient," or "differential coefficient with respect to x," or X', is represented by the tangent of an angle on the paper.

In fig. 6 the x's are again represented by horizontal lines, and the differential coefficients are also represented by vertical lines, while the integral is represented by an area. The known values of X'for successive values of x are plotted upwards co-ordinate with the x's, and a curve is thus obtained. The area of the vertical strip, bounded at the top by the curve, at its sides by two vertical straight lines dx apart, and at its foot by the axis of x, equals the product X'dx, where X' is the mean value throughout the length &c. The successive strips thus formed are contiguous to each other, so that the sum of them, that is, the integral X'dx, is the total area underneath the curve and above the axis of x included between the two verticals at x1 and x, the limiting values of x.

Here, again, an error would be made if, instead of taking the mean value of X' for each length Sx, there were taken either its value at the beginning of dx or that at the end of dx.

If for each

Sx the value of X' at its beginning were used, the total area, or integral, obtained would be too small by the sum of the small triangular areas rafined as in the lower parts of the rectangles in fig. 6. If, on the other hand, the value at the end of each Sx were used, the calculated integral would be greater than the true integral by the sum of the small triangular areas rafined as in the upper parts of the rectangles in fig. 6. This is clearly true whether the da's be uniform in size or of differing sizes.

49. Diminishing Error.-Not only does the size of each such error for each individual de diminish indefinitely as dx is made smaller, but their sum also diminishes indefinitely in spite of their number being indefinitely increased. Although this is not really a

proposition of any special theoretical importance, since it is always the average, and not either the initial or final values of X' that are to be used, still it is interesting and customary to prove it to be true. The neatest proof is that given by Newton, and is explained by fig. 6. For the purposes of the proof all the Sa's are assumed to be of one uniform size, and, therefore, each of them equal to the last before X2 x, is reached. This being so, all the above-mentioned triangular area-errors are transferred horizontally so as to lie between the vertical X2 and (x2 - 8x). The sum of each set of errors is now easily seen to be equal to about (X'2-X')dx, the sum of both sets taken together being exactly equal to this last strip. Now the height (X2-X') of this strip has a definite finite known value; and, therefore, the area (X'2-X')dx diminishes always in size as Sx is made smaller.

It must be clearly understood that no such error, large or small, occurs if the X's employed are the true average heights of the narrow strips. The average height of the strip is found at the middle of its breadth when the upper boundary is straight, and deviates from the middle by only a minute distance, if the upper boundary be very nearly straight, as it does become when its length is made very short.

50. Integration through Infinite Gradient.-A simple consideration of the vertical cliff below J, in fig. 1, where the gradient is infinite, will show that, while the process of integration written

dh

dl

. dl is possible up to the base-point of the cliff, and also possible from the top edge of the same onwards, it is impossible throughout the height of the cliff, simply because it has no meaning. Here neither the gradient nor the increment 87 (or Sx) has any existence. Mathematicians so constantly speak of the infinitely great and the infinitely small, or infinity and zero, that probably many novices in mathematical reasoning fall into the habit of imagining that there are really such things. But neither has any sort of existence except as symbols, and in this case the symbols correspond to no physical fact, and to no mental conception. Throughout the vertical cliff there is no horizontal distance, i.e., there is no 87; and there is no gradient, because a gradient is the ratio between the vertical and horizontal projections of a sloping length, and since there is here no horizontal projection, there can be no ratio between the two.

The phrases "infinity" and "zero" are merely word-symbols to indicate the directions in which very large things and very small things vanish beyond the range of our perceptions, as they grow larger and smaller.