16 VALUE OF THE TERM SCIENCE. borne in mind is, that from the whole history of Chemistry no reason can be elicited why an alkali should be soluble in water rather than insoluble, or soluble in alcohol rather than insoluble; why it should combine with oils or acids rather than resist combination with them; why it should change vegetable blues to green, and yellows to brown, rather than to any other colour. In the conception of properties, as belonging to the alkalies, opposed to all those just enumerated, there is nothing contradictory. In short, there is no reason why any peculiar property of an alkali, so far as the human faculties can comprehend, should not, in the arrangement of nature, have been the opposite of what it actually is. And the same may be said of all those laws and properties in nature which are discovered solely by observation. On the contrary, when the several steps are considered by which an equality is proved between the square of the hypotenuse in a right-angled triangle, and the sum of the squares on its two sides, there is not discoverable, in the whole course of the demonstration, any single truth, the opposite of which does not involve a contradiction, so that, independently of any observation, the human mind is, by its very nature and constitution, compelled to extend to them an absolute and unconditional belief. The square described on the hypotenuse being cut by a straight FIG. 1. line in such manner as divides it into The proof of the theorem just referred to, may readily be understood even by one unversed in the elements of geometry. With the meaning of parallel lines every one is familiar. Here are three pairs of parallel lines; one pair running from side to side, and two pairs between them, forming two parallelograms of rectilinear figures, the opposite sides of which are parallel. These two parallelograms stand upon the same base, and lie between the same parallels; and when this is FIG. 2. the case parallelograms are equal that is, the area of the more EVIDENCE OF MATHEMATICAL TRUTH INTUITIVE. 17 upright of these two figures is equal to the area of the more slanting figure. And the truth of this will appear at once, by considering how the whole figure, composed of the two parallelograms taken together, is made up. If from this whole figure the more upright of the two parallelograms be taken, a triangle remains; and if from the whole figure the more oblique parallelogram be taken, another triangle remains. But these two triangles are equal, their corresponding sides and angles being equal; hence the parallelogram which remains, after one of these triangles is taken away, must be equal to the parallelogram which remains after the other triangle is taken away. Such, then, is the proof of the proposition, that parallelograms between the same parallels, and standing on the same or an equal base, are of the same area; and as every parallelogram is divisible into two equal triangles by its diagonal, it follows that triangles standing on the same base, and between the same parallels, are of the same area. Let us now return to the figure on the preceding page, representing the squares on the three sides of a right-angled triangle. In this figure there is a triangle standing on the same base, and between the same parallels as the square on the left-hand side of the triangle, and there is a triangle standing on the same base, and between the same parallels as the larger of the two parallelograms into which the square of the hypotenuse is divided; but these two triangles are equal, owing to the equality of two sides, and the contained angle; hence the square, which is equal to twice the area of one of these equal triangles, is equal to the parallelogram, which is equal to twice the area of the other triangle. And by the same mode of reasoning, the square on the right-hand side of the triangle is proved to be equal to the lesser of the two parallelograms into which the square of the hypotenuse has been divided. But in the whole range of Geometry the proposition holds good, that every stage of the proof is a truth, the opposite of which involves a contradiction; and therefore, that it is itself a necessary article of belief. In short, it is incontrovertible that mathematical truths are necessary truths. Geometricians use various ways of convincing us of this: where two figures are necessarily equal, as a consequence of certain parts in one being known to be equal to corresponding parts in the other, the method of superposition is frequently employed; that is, we are required to fancy one figure placed upon the other, and then, mentally, to bring about their perfect adaptation: the parts, previously known to be the same in both, being properly adjusted, the other parts, by this method, are shown to be necessarily coincident. There is, however, nothing of a mechanical or experimental character in this process: the figures are not bodily transported from one place to another; the whole is a 18 OBJECTS OF MATHEMATICAL SCIENCE. purely mental operation; and it is the mind, not the eye, that sees the complete adaptation of the two. Some superficial thinkers cavil at the peculiar character assigned to mathematical science, by reference to the very proposition above adduced; saying that the fact as to the equality of the squares, was discovered by observation, and the demonstration afterwards invented; as is proved, they further say, by the tradition, that Pythagoras sacrificed a hecatomb in gratitude to the gods, for having inspired him with its discovery. Thence, it may be supposed, they would infer, that all mathematical knowledge is founded on observation, and not on intuitive convictions of the human mind. It is evident, however, that many truths, susceptible of a mathematical demonstration, like that respecting the squares on the sides of a right-angled triangle, are discoverable by observation; and doubtless, in the early progress of geometry, this method was much employed to discover the course to be adopted for the extension of this branch of knowledge. But had geometry, or any other part of mathematics, been confined to this method of investigation, would it ever have attained the rank of being the handmaid of inductive science-the very means by which observation has been made capable of deciphering the system of the universe? The distinction between mathematical truth and inductive science, so clearly pointed out by the contrast between the properties of the alkalies, and the remarkable properties of the right-angled triangle above referred to, is irrefutable. Magnitude. We have not hitherto referred to the great object which mathematical science has in view, namely, to supply a measure by which all magnitudes may be rendered commensurable. A few words will give the steps by which this is accomplished in a sufficiently clear light. By the propositions readily reducible to the truth, before referred to, that two triangles are equal, if their corresponding angles and corresponding sides be equal, any two rectilineal figures, however 、 dissimilar, may be proved to be equal if they really be equal, or unequal if they be unequal. And this may be described as the first great step in Mathematical Science; because, by means of the equivalence of triangles, all rectilineal figures are rendered commensurable. The next step in Mathematics is to find the measure of figures bounded by curved lines. For example, to find the area of a circle in rectilineal measure. The attempts to find the area of a circle in rectilineal measure gave rise to the proof by the method of " exhaustions," as it is termed. OBJECTS OF MATHEMATICAL SCIENCE. 19 The area of a circle is a quantity intermediate between the area of a polygon circumscribing the circle, FIG. 3 exterior polygon can never fall short of the area of the circle, nor that of the interior polygon exceed the area of the circle. Thus, as the sides of these polygons may be increased without any limit, the difference between the area of the exterior polygon and the area of the interior polygon is continually becoming less and less, or continually approaching, without reaching, to nothing; and though the rectilineal polygon cannot be made an exact measure of the curvilineal circle, yet it can be made to approach to its measure with any required degree of nearness. It may be remarked here, also, that this operation enables the unlearned reader to understand what is meant when it is said that unity divided infinitely = 0. It was another step in Mathematics when the area of curvilineal figures came to be expressed exactly by the areas of rectilineal figures. What are called the "lunes" of Hippocrates, known to the ancients, afforded one of the earliest examples of this coincidence. To exhibit this property, a right-angled triangle is inscribed in a semicircle, and a semicircle described on its base and its perpendicular. The portions of the two last semicircles which lie without the original semicircle, are found to be equal to the area of the triangle. The following is the kind of proof on which this proposition rests. It is found that if semicircles are de scribed on the three sides of a rightangled triangle, the area of that described on the hypotenuse is equal to the joint areas of the semicircle on the base and that on the perpendicular. But the greater semicircle in the annexed figure consists of the rightangled triangle and the two arches of FIG. 4. that semicircle cut off by the sides of the triangle, and the joint areas of the two lesser semicircles consist of the two lunar spaces cut off by the greater semicircle and the two arches of that great semi 20 MEASURE OF CURVILINEAR MAGNITUDES. circle just mentioned; hence, if from each of these two equal quantities, the common quantity in both, namely, the arches of the great semicircle cut off by the sides of the triangle, be taken away, there remains on the one hand the triangle, and on the other the lunar spaces of the lesser semicircles, taken together, equal to each other. The propositions, on which the proof of this correspondence in equality depend, are easily understood. a The circumference of a circle is proportional to its diameter proposition which may easily be shown to be a necessary consequence of the geometrical definition of proportion. It is not, however, so obvious that the area of one circle is to the area of another circle, as the square of the diameter of the first circle to the square of the FIG. 5. diameter of the second circle. It is, however, a very important proposition, for, if a person supposed that the areas of circles are simply proportionate to their diameters, he might commit many serious errors. For ex ample, if he wished a tube, as a gas-tube, twice the capacity of another tube, and desired it to be made of equal length, but twice the diameter, it would turn out to have four times the capacity; for the square of a line eight inches long consists of sixty-four square inches, while that of a line four inches long consists of only sixteen square inches. That the areas of circles are not to one another as their diameters, FIG. 6. is a truth of which the learner may easily satisfy himself without any knowledge of Geometry; thus: let a circle be described with any diameter, and within it let two circles be described, with the diameter of each only half that of the outer circle; then if a circle, with double the diameter of another, were no more than double that of the other in area or surface, it is plain that the two inner circles would just fill up the outer, which is at once seen to be impossible. It is, however, worthy of remark, that the circumference of the outer circle would be exactly equal to the two circumferences of the inner circles, which is only one among the many interesting and unexpected truths that Geometry presents. But the great progress made in this part of Mathematics has arisen from the investigation of the areas produced by the higher order of curves, as of the conic sections, exemplified in the ancient |