VALUE OF THE TERM SCIENCE.

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systems of knowledge which are built upon the perceptions of sense, variously grouped into a whole, because of the agreement of the members of each group in one mode of presenting themselves. But vague as is the term science, it is too firmly rooted to be rejected.

Geometry. When any part of Mathematics, for example, Geometry is compared with some one of the Inductive Sciences, such as Chemistry, it is discovered how loosely the term science must be used to apply equally to both. For this purpose, we select for contrast the properties of the alkalies, on the one hand, and on the other the remarkable property of the right-angled triangle, that the square of its hypotenuse is equivalent to the sum of the squares of the two sides. The alkalies-that is, the pure caustic alkalies —are freely soluble in water and in alcohol; each saturates its own proportion of every known acid; and were a new acid discovered, it would only be necessary to ascertain how much of it is required to saturate a given quantity of one of the alkalies, to pronounce how much of each of the others that same quantity would saturate; the alkalies, besides, form soaps with oils; they change vegetable blue colours to green, and yellows to brown. By means of these properties, the chemist is able to detect the presence of any pure alkali in his analysis; and such is one of the great objects which the science of Chemistry has in view. But the point which we wish chiefly to be 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 line in such manner as divides it into two distinct parallelograms, it is at once shown by the undeniable proposition, that if two equals have each an equal quantity added to them, the sums are equal; and then by the undeniable proposition that the doubles of equals are equal to one another-that each of the two divisions of the square on the hypotenuse is equal to one of the squares on the two sides of the triangle.

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 parallelo

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EVIDENCE OF MATHEMATICAL TRUTH INTUITIVE.

grams or 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 the case parallelograms are equal-that is, the area of the more 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 samé 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 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.

OBJECTS OF MATHEMATICAL SCIENCE.

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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.

The area of a circle is a quantity intermediate between the area of a polygon circumscribing the circle, and that of a similar polygon inscribed within the circle. If the number of sides in each of these polygons be successively increased, the area of the interior polygon is continually augmented, while the area of the exterior polygon is continually diminished,-plainly, however, on this condition, that though the area of each continually approaches nearer and nearer to the area of the circle, that of the 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

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THE MEASURE OF CURVILINEAR MAGNITUDES BY RECTILINEAR.

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 described 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 right-angled triangle and the two arches of 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 semicircle 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.

The circumference of a circle is proportional to its diameter a 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 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 example, if he wished a tube, as a gas-tube, twice the capacity of another tube, and desired it to be made of

D

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, 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,

THE EVIDENCE OF THE PROPERTIES OF NUMBER INTUITIVE.

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exemplified in the ancient discovery that a parabola is equal to two-thirds of its circumscribing parallelogram.

But it would be superfluous to carry these illustrations further, since it already sufficiently appears what is the proper object of Mathematics, and that the evidence employed in this Science uniformly consists of propositions, the reverse of which, according to the constitution of the human mind, involves a contradiction.

Number.-Our observations have been confined hitherto to what relates to magnitude; but the doctrine of number is in no respect different. That 2 and 2 make 4, and that 2 taken from 4 leave 2, are unquestionably intuitive truths-they must be believed; they are necessary truths, because the opposite propositions involve a contradiction. But the truth that 10 times 10 make 100, rests on the same kind of evidence. One repeated a hundred times makes 100. Observation is not required to prove 10 times 10 to be 100; it is merely required to discover if what is called 100 be 100, If, in the primitive state of our race, one man, on giving another figs or dates, held up the fingers of both hands ten times, he who received them would count them, not to ascertain if 10 times 10 were 100, but to discover if he who gave the fruit had spoken truly as to the number.

Mathematical Evidence.-All Arithmetic, then, rests on the same evidence—all its truths are necessary; and the same may be said of Algebra, Logarithms, and the Differential Calculus. Algebra may be described as Arithmetic carried on by symbols; so that the kind of operation is constantly indicated, but not actually performed till the relation between the given quantities and the quantity sought, be reduced to its simplest possible form. Logarithms depend on what seems a singular property of numbers; yet that property is as certainly deducible from necessary truths as any truth in Mathematics. If two series of numbers stand respectively in Geometrical and Arithmetical ratio, it is found that the product of any two numbers in the Geometrical series may be found by adding the corresponding numbers in the Arithmetical series, and then taking the number in the Geometrical series which stands opposite: and this is the product sought.

Logarithms.—The most difficult and complicated arithmetical operations may be performed with ease and expedition by means of Logarithmic tables; and thus multiplication is reduced to addition, division to subtraction, evolution to multiplication, and the troublesome process of involution, or the extraction of roots, to simple division. Astronomy owes much of its pre-eminence, as an exact science, to the discovery of Logarithms, as, without their aid, it would have been almost impossible to have made the calculations necessary to confirm its laws. The astronomer reduces his algebraical formulæ to a form adapted for logarithmic computation; and his assistants, by the simplest rules of arithmetic, are thus enabled to compile the Nautical Almanac, without which the commerce of our great nation would be nearly destroyed-the Nautical Almanac and a table of logarithms being as essential to the mariner as his chart and compass.

Proportion. To exhibit a tithe of the uses to which the sciences of quantity and number can be applied, would fill a volume. Still the only practical use of these important sciences, is the measurement of quantities before unknown. The great instrument in all the departments of abstract science is proportion; thoroughly to understand which is to possess an instrument of knowledge applicable to almost every situation in life, When Thales of Miletus travelled into Egypt, 600 years before Christ, and saw the Great Pyramid, he was curious to determine its height, which hitherto it had been