Again it is mathematically demonstrable, that a straight line, the asymptote of a hyperbola, may eternally approach the curve of the hyperbola and never meet it. But no axiom can be plainer than that if two lines continually approach each other they must at length meet. Here is a demonstration contradicting an axiom; and no man has ever yet shown the possibilities of reconciling them, nor yet of denying either side of the contradiction.

Again it is a fundamental axiom, contained in the definition of a circle, that it must have a center; but the nonexistence of this center is mathematically demonstrable, as follows: Let the diameter of the circle be bisected into two equal parts; the center must be in one, or the other, of these parts, or between them. It can not be in one of these parts, for they are equal; and, therefore, if it is in the one, it must also be in the other, and thus the circle would have two centers, which is absurd. Neither can it be between them, for they are in contact. Therefore the center must be a point, destitute of extension, something which does not occupy or exist in space. But as all existences exist in space, and this supposed center does not, it can not be an existence; therefore it is a non-existence.

In like manner it has been mathematically demonstrated,* that motion, or any change in the rate of progress in a moving body, is impossible; because in passing from any one degree of rapidity to another, all the intermediate degrees must be passed through. As when a train of cars moving four miles an hour strikes a train at rest, the resulting instantaneous motion is two miles an hour; and the first train must therefore be moving at the rate of four, and at the rate of two miles an hour at the same time, which is impossible. And so the ancients demonstrated the impossibility of motion.

* Journal of Speculative Philososphy, I. 20.

Thus the non-existence of the most undeniable truths, and the impossibilities of the most common facts are mathematically demonstrable; and the proper refutation of such reasoning is, not the scientific, but the common sensible; as when Plato refuted the demonstration of the impossibility of motion, by getting up and walking across the floor. In the hyperbola we have the mathematical demonstration of the error of an axiom. In the infinite inch we behold an ab

surdity mathematically demonstrated So that it appears we can give mathematical demonstration in support of untruth, impossibilities and absurdities; and our reason can not discover the error of the reasoning! Alas, for poor humanity, if an endless destiny depended upon such scientific certainty! Yet mathematical reasoning about abstract truth is universally conceded to be less liable to error than any other form of scientific analysis This line, then, is too short to fathom the ocean of destiny; too weak to bear inferences from even the facts of common life.

Attempts have indeed been made to apply mathematics to the facts of life in what is called the doctrine of chances.

By this kind of calculation it can be shown, that the chances were a thousand millions to one that you and I should never have been born. Yet here we are.

But when we begin to apply mathematics to the affairs of every-day life, we immediately multiply our chances of error by the number and complexity of these facts. The proper field of mathematics is that of magnitude and numbers. But very few subjects are capable of a mathematical demonstration. No fact whatever which depends on the will of God or man can be so proved. For mathematical demonstration is founded on necessary and eternal relations, and admits of no contingencies in its premises. The mathematician may demonstrate the size and properties of a triangle, but he can not demonstrate the continuance of any actual triangle for one hour, or one minute, after his dem

onstration. And if he could, how many of my most important affairs can I submit to the multiplication table, or lay off in squares and triangles? It deals with purely ideal figures, which never did or could exist. There is not a mathematical line-length without breadth—in the universe. When we come to the application of mathematics, we are met at once by the fact that there are no mathematical figures in nature. It is true we speak of the orbits of the planets as elliptical or circular, but it is only in a general way, as we speak of a circular saw, the outline of its teeth being regularity itself compared with the perturbations of the planets. We speak of the earth as a spheroid, but it is a spheroid pitted with hollows as deep as the ocean, and crusted with irregular protuberances as vast as the Himalaya and the Andes. in every conceivable irregularity of form. Its seas, coasts, and rivers follow no straight lines nor geometrical curves. There is not an acre of absolutely level ground on the face of the earth; and even its waters will pile themselves up in waves, or dash into breakers, rather than remain perfectly level for a single hour. Its minuter formations present the same regular irregularity of form. Even the crystals, which approach the nearest of any natural productions to mathematical figures, break with compound irregular fractures at their bases of attachment. The surface of the pearl is proportionally rougher than the surface of the earth, and the dew-drop is not more spherical than a pear. As nature then gives no mathematical figures, mathematical measurements of such figures can be only approximately applied to natural objects.

The utter absence of any regularity, or assimilation to the spheroidal figure, either in meridianal, equatorial, or parallel lines, mountain ranges, sea beaches, or courses of rivers, is fatal to mathematical accuracy in the more extended geographical measurements. It is only by taking the mean of a great many measurements that an approxi

mate accuracy can be obtained. Where this is not possible, as in the case of the measurements of high mountains, the truth remains undetermined by hundreds of feet; or, as in the case of the earth's spheroidal axis, Bessel's measurement differs from Newton's, by fully eleven miles.* The smaller measures are proportionately as inaccurate. No field, hill, or lake, has an absolute mathematical figure; but its outline is composed of an infinite multitude of irregular curves too minute for man's vision to discover, and too numerous for his intellect to estimate. No natural figure was ever measured with absolute accuracy.

All the resources of mathematical science were employed by the constructors of the French Metric System; but the progress of science in seventy years has shown that every element of their calculations was erroneous. They tried to measure a quadrant of the earth's circumference, supposing the meridian to be circular; but Schubert has shown that that is far from being the case; and that no two meridians are alike; and Sir John Herschel, and the best geologists, show cause to believe that the form of the globe is constantly changing; so that the ancient Egyptians acted wisely in selecting the axis of the earth's rotation, which is invariable, and not the changing surface of the earth, as their standard of measure

The Astronomer Royal, Piazzi Smyth, thus enumerates the errors of practice, which they added to those of their erroneous theory: "Their trigonometrical survey for their meter length has been found erroneous, so that their meter is no longer sensibly a meter; and their standard temperature of 0° centigrade is upset one way for the length of their scale, and another way for the density of the water employed; and their mode of computing the temperature correction is proved erroneous; and their favorite natural * Humboldt, Cosmos, Vol. I. p. 7, 156.

reference of a quadrant of the earth is not found a scientific feature capable of serving the purpose they have been employing it for; and even their own sons show some dislike to adopt it fully, and adhere to as much of the ancient system as they can."*

But coming down to more practical and every-day calculations, in which money is invested, how very erroneous are the calculations of our best engineers, and how fatal their results. Nineteen serious errors were discovered in an edition of Taylor's Logarithms, printed in 1796; some of which might have led to the most dangerous results in calculating a ship's place, and were current for thirty-six years. In 1832 the Nautical Almanac published a correction which was itself erroneous by one second, and a new correction was necessary the next year. But in making this correction a new error was committed of ten degrees † Who knows how many ships were run ashore by that error?

Nor can our American mathematicians boast of superior infallibility to the French or British. In computing the experiments which were made at Lowell (for a new turbine wheel), it was found that when the gate was fully open, the quantity of water discharged through the guides was seventy per cent. of the theoretical discharge. (An error of thirty per cent.) The effect of the wheel during these experiments was eighty-one and a half per cent. of the power expended, but when the gate was half open the effect was sixty-seven per cent. of the power, while the discharge. through the guides eleven per cent. more than the theoretical discharge. But when the opening of the gate was still further reduced to one-fourth of the full opening, the effect was also reduced to forty-five per cent of the power, while the discharging velocity was raised to forty-nine per

* Our Inheritance in the Great Pyramid, 356. † Annual of Scientific Discovery, 1852.