EVIDENCE OF THE PROPERTIES OF NUMBER.

21

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 algebrai

22

MEASUREMENT OF INACCESSIBLE HEIGHTS.

cal 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 deemed impossible to ascertain. Observing the shadow of the pyramid as the sun shone upon it, stretching far in the opposite direction, he struck his staff upright in the sand; and finding the shadow which it cast to be exactly its own length, he rightly concluded that the shadow, measured from the middle of the base of the pyramid, must equal in length the height of the pyramid. He paced the shadow, and found its length to be 270 paces, or about 500 English feet. Pliny, who relates this anecdote (lib. xxxvi. 17), expressly says that Thales measured the shorter shadow at the time when it was of the same length as the staff.

But although equality in length of the shadow and the body may be allowed to be necessary for the discovery of this mode of mensu

MEASUREMENT OF HEIGHTS WITHOUT PROPORTION.

23

ration, it would quickly appear without any necessity for experiment, that whatever relation the shadow bore to the staff, the same relation of magnitude would the shadow bear to the height of the pyramid. The three things requisite are, the measure of the shadow of the staff, the measure of the shadow of the pyramid, and the measure of the staff itself. But, to solve this more complex problem, the knowledge of proportion is necessary: namely, that when of four numbers the first two bear the same analogy to each other as the last two to each other, the first of the four, multiplied by the last of the four, is equal to the second multiplied by the third; or, as it is usually expressed, the product of the extremes is equal to the product of the means:-or 4: 16 :: 20: 80-that is, 4 is to 16, as 20 to 80; but the product of the extremes, 4 and 80, is 320; and the product of the means, or middle numbers, 16 and 20, is also 320. But when three numbers are known, and a fourth is sought in the same relation to the third which the second holds to the first, it is plain that the product of the means can be obtained, and that that product being also the product of the extremes when both these come to be known, and being divided by the first extreme, the second extreme will be obtained: that is, if in the above formula 16 and 20 be multiplied together, and the product divided by 4, the fourth number, the second of the two extremes, or 80, will be obtained.

And this rule of proportion prevails throughout the whole range of the sciences of magnitude and number, In every kind of measurement proportion plays its part, with the exception of that which is of the rudest kind. In the measurement of the height of the

FIG. 8.

A

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Great Pyramid by Thales, the idea of proportion is involved, although hardly brought out into relief. We will cite another example of the measurement of a height without distinct reference to proportion. The height of a tower or pillar-no matter how high-which

24

MEASUREMENT OF HEIGHTS WITHOUT PROPORTION.

stands on a level plain, and the foot of which is accessible, can be measured as soon as men have discovered that in a right-angled triangle, the sides of which are equal, each of the other two angles is equal to half a right angle, and the perpendicular equal to half the hypotenuse. If the perpendicular line in a right-angled triangle represent a tower, it is evident that its height is equal to half the hypotenuse, or side opposite to the right angle at A. Thus, if a person setting out from the foot of the tower pace the distance to the point at which the top of the tower is seen at an elevation of 45°, or half a right angle, the number of paces he has taken indicates the height of the tower.

Trigonometry. The usual mode of determining heights is by the rules of Trigonometry, without any necessity for the angle of elevation being of a particular number of degrees. When a tower is accessible, the angle BCA is measured, and the

1.00

B

FIG. 9.

FIG. 10.

C

base of the triangle CB; the angle at B is known, being a right angle, and the angle at A is found by subtracting the angle at C from 90° or a right angle; because since the three angles of every triangle are together equal to two right angles, the angles at C and A are together equal to one right angle.

When the foot of the tower is inaccessible, the angle GFE is measured, then the space FD and the angle FDE; the angle EFD is found by subtracting GFE from two right angles, since every straight line falling on another straight line forms with it two angles, together equal to two right angles. But when the angles D and F in the triangle EDF are known, the angle at E is easily found by subtracting the sum of the angles D and F from two right angles. But as a general rule in Trigonometry, when out of the three sides and three angles of a triangle, any three, except the three angles, being given, the remaining three can be determined. Hence the length of the line AB in the triangle ACB, or the height of the tower, can be so dis

LAWS OF MOTION..

25

covered; and in the triangle FDE the length of EF can be discovered, as preliminary to the same steps.

Motion. The laws of motion, which make up so important a part of Natural Philosophy, stand at once on a different footing from mathematical truth, and from the principle of gravitation. It is common to enumerate three laws of motion. The first is, that a body under the action of no external force will remain at rest, or move uniformly in a straight line. The second, that when a force acts upon a body in motion, the change of motion in magnitude and direction is the same as if the force acted upon the body at rest. The third law of motion is, that when pressure communicates motion to a body, the momentum generated in a given short time is proportional to the pressure, or, as given by Newton in a more general form, action and reaction are equal and opposite.

In order to form a correct notion of these laws, we must have definite ideas of bulk, force, velocity, motion, and pressure, as well as the modes of measuring them. Newton defines the mass of a body to be the product of its density and its volume; and he determines the mass by its weight, because he found, by most accurate experiments with pendulums, that the mass is proportional to the weight. We see that all bodies placed above the earth's surface have a tendency to fall, and exert a force upon whatever support prevents them from falling; this force we term pressure, and the measure of this pressure is weight-bodies being said to be of equal weight, if they produce equal pressure on their support; consequently weight is a measure of the earth's attraction for heavy bodies; but, in assuming weight to be a measure of mass, or the quantity of matter contained in a body of given volume, we clearly assume that the earth's attraction is the same for all kinds of matter; and that a cubic inch of gold weighs more than a cubic inch of copper, because the former contains more particles of matter than the latter, and not because the earth has a more powerful attraction for gold than copper

an assumption abundantly confirmed by experiment. Hence weight becomes a measure of pressure, and consequently of force producing pressure. We can also estimate force, in another way, without reference to mass, pressure, or weight. According to the first law of motion, a body can only move by the action of some external force; now, the space through which it passes in a given time will afford us a measure of its velocity, which is only a term for the quickness or slowness of its motion; and the velocity acquired in any given time will afford us a measure of the force which produces the motion of the body. Neglecting the resistance of the air, it is found that all heavy bodies, how different soever in weight, fall through the same space, and acquire the same velocity at the end of any given interval of time. It is clear, therefore, that the measure