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science; tradition says that it was discovered by Pythagoras about 2500 years ago, and that he offered 100 oxen in sacrifice to shew his gratitude: In any right-angled triangle the square described on the hypotenuse is equal to the sum of the squares described on the sides.

33. In the Mensuration it is explained how the truth of the preceding statement is rendered visibly self-evident. It is also shewn that when we know the lengths of the two sides we can deduce that of the hypotenuse; and that when we know the lengths of the hypotenuse and of one side, we can deduce that of the other side. It is obvious that the hypotenuse is longer than either of the sides; because the square on the hypotenuse is greater than the square on either side.

34. The important notion of proportion makes its appearance in Geometry. Thus, to take a single example, let there be two triangles ABC and DEF, such that the angle A is equal to the angle D, the angle B equal to the angle E, and the angle C equal to the angle F; then the corre sponding sides are in proportion.

A A

That is to say, whatever may be the proportion of DE to AB, the proportion of EF to BC is the same, and so is the proportion of FD to CA. Thus, for example, if DE is twice AB then EF is twice BC, and FD is twice CA.

35. As a good example of proportion we may take the case of the relation between the heights of two objects and the lengths of their shadows in sunlight. It will be plain to a reader who has a little acquaintance with Optics that the two heights are in the same proportion as the two corresponding lengths.

36. The two triangles in Art. 34 are said to be similar. And in general two plane figures are said to be similar when one is exactly a copy of the other on a larger or smaller scale. It is an important property of similar figures that their areas are in the same proportion as the squares of the numbers which denote corresponding lengths. For instance if EF is 2 times BC then the area of the triangle DFE is to that of the triangle ABC in the same proportion as the square of 2 to the square of 1, that is in the proportion of 4 to 1.

37. In like manner when one solid is exactly a copy of another on a larger or smaller scale the solids are said to be similar. It is an important property of similar solids that their volumes are in the same proportion as the cubes of the numbers which denote corresponding lengths. Thus if a brick were made 4 inches long, 2 inches broad, and 11 inches thick, it would be similar to the ordinary brick of Art. 21. And as the length of an edge of the ordinary brick is 2 times the corresponding length on the smaller brick, the volume of the ordinary brick is to the volume of the smaller in the same proportion as the cube of 2 to the cube of 1, that is in the proportion of 8 to 1.

38.

We can always draw straight lines the lengths of which shall be in any proportion we please, and thus by the means of straight lines we may often conveniently bring such a proportion before the eye. Take for example 14 pounds and 112 pounds; the latter is eight times the former, and so the proportion may be represented to the eye if we draw one straight line of any length we please and another straight line eight times as long.

39. Philosophers have occasionally speculated on the possibility of constructing some universal language which should serve as a medium of intercourse between all mankind. Up to the present time the signs and diagrams of mathematics seem the nearest approach to the realization of such a scheme; for by their aid, almost without any vocabulary, truths may be rendered intelligible to nations of the most different languages. And even a still wider range

has been suggested for the prevalence of this mode of communication. "We can conceive occurrences which would give us evidence that the Moon, as well as the Earth, contains geometers. If we were to see, on the face of the full moon, a figure gradually becoming visible, representing a right-angled triangle with a square constructed on each of its three sides as a base; we should regard it as the work of intelligent creatures there, who might be thus making a signal to the inhabitants of the earth, that they possessed such knowledge, and were desirous of making known to their nearest neighbours in the solar system, their existence and their speculations." Plurality of Worlds, Chapter IV.

40. In asking the beginner to give his attention to the work on Natural Philosophy now put into his hands, it will be well to remind him that the knowledge which he gains from the book should be confirmed and extended by carefully watching the phenomena which spontaneously offer themselves to his observation, and also by attending good experimental lectures if such be within his reach. Some attempt might be made to supersede the advantage of external observation and experiment by elaborate drawings, but it is difficult to make these easily intelligible without familiarity with the objects they represent, and after such familiarity they become superfluous. While it may be readily admitted that books on Natural Philosophy alone do not make a sufficient impression on the mind and memory, it is equally certain that a book in which the principles are recorded and explained, is a necessary accompaniment to the oral and visible teaching of the lecture room. It must not be forgotten that in the course of life books are always and everywhere accessible, but lectures by no means so certainly; hence too much stress cannot be laid on the importance of early acquiring the habit of learning from books. In these days of diffused knowledge it is curious to observe how many persons of respectable education are practically unable to read; though they may be fluent in conversation and quick to appreciate what is made audible or tangible, they have never accustomed themselves to apply with close attention to the silent and unobtrusive teaching of the printed page.

41. It has been the singular honour of some elementary books intended mainly for youth that they have fallen under the notice of persons of maturer power, and have thus indirectly influenced the history of science. Thus it has been stated that Mrs Marcet's Conversations on Chemistry "first opened out to Faraday's mind that field of science in which he became so illustrious, and at the height of his fame he always mentioned Mrs Marcet with deep reverence." (Mrs Somerville's Personal Recollections..., page 114.) The same book had the honour of Dr Whewell's attention; he read it and made a short analysis of it in 1817. A sentence in Mrs Somerville's Connection of the Physical Sciences incited a living astronomer to undertake the laborious investigation which finally enabled him to ascertain the existence of the planet Neptune, then unknown.

II. VARIOUS BRANCHES OF KNOWLEDGE.

42. Many eminent philosophers have turned their attention to the subject of the classification of the various branches of knowledge, and though no solution of the difficult problem has been obtained which is entirely satisfactory, yet the attempts have been interesting and instructive. We shall not give here any elaborate discussion of the subject, but a few remarks will be advantageous, as they will furnish a general idea of the range of the present work.

43. It will be sufficient for our purpose to consider that there are five main branches of knowledge; these may be called Mathematical, I hysical, Chemical, Vital and Mental.

44. The Mathematical sciences relate to number and to figure. They have as their foundation Arithmetic and Geometry. They are sometimes called abstract sciences, being to a great extent independent of all that takes place in the world around us, and derived by the human mind

from its own resources. These sciences have been cultivated from the days of the ancient Greeks to our own, and as, from their nature, whatever has been once established in them remains as a permanent truth, an enormous mass of striking and valuable results has been accumulated by the labour of successive generations. As we have already said, the amount of mathematical knowledge assumed for the purposes of the present work is very slight.

45. The Physical sciences are often called Natural Philosophy. Such sciences might have originally included the knowledge of everything which the world of Nature contains; but at present the term is somewhat restricted in its application. Natural Philosophy now may be said to include a group of sciences which has grown up round Astronomy, the oldest and most perfect of them all. Astronomy at first involved only observations of the situations of the heavenly bodies, and predictions of their future course from the records of the past; but Newton by his theory of gravity extended the subject and deduced the motions of the moon and the planets from one general law. Then the whole science of Mechanics in its widest sense was gradually formed; this treats of the counexion between force and the motion which it produces or changes or arrests, and it has different names according as it relates mainly to motion or to rest, to solid or to fluid bodies. With Astronomy is naturally connected the science which treats on Light, the medium by which so much of our knowledge of the skies is obtained; and Navigation which is closely connected with Astronomy introduces Magnetism in virtue of the Mariner's Compass. Light may be said to draw with it the kindred subject of Heat, and Magnetism all the train of sciences which in modern times have sprung from this and Electricity. The progress of every science and of every part of a science resembles that of Astronomy; it is traced back to more simple and more general principles as its origin, and carried forward to more numerous and more varied applications and extensions.

46. The Chemical sciences take their rise from the fact that there is more than one kind of substance in nature. Had there been only one kind, what is called Chemistry

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